ABSTRACT. We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems. The key idea is to represent each conservation law or dissipation inequality by means of an associated test function; we introduce auxiliary variables representing the projection of these test functions onto a discrete test set, and modify the equation to use these new variables.

We demonstrate these ideas by their application to the several problems, including Hamiltonian and GENERIC ODEs, the Benjamin-Bona-Mahony equation, and the Navier-Stokes equations; we devise a time discretization of the Eulerian formulation of the compressible Navier-Stokes equations that conserves mass, momentum, and energy, and provably dissipates entropy. In several cases preserving conservation and/or dissipation structure appears to offer considerable qualitative advantages over preserving other important structures, like symplecticity.

ABSTRACT. Understanding, quantifying and controlling transport and mixing processes are central in the study of fluid flows. Many different Lagrangian approaches have been proposed for detecting organizing flow structures. We review a simple network-based framework for the extraction of coherent sets directly from tracer trajectories. We combine this with diffusion maps and aspects of deterministic particle methods to study mixing processes. We demonstrate our approach in a number of example systems, including trajectory data from experimental time-resolved particle tracking (4D-PTV) in a lab-scale stirred tank reactor.

ABSTRACT. In this talk, I will present a model for continuous-time opinion dynamics on an evolving network. In this model, individual’s opinions evolve continuously in time according to a Hegselmann-Krause model, but where the interaction between agents is mediated by a network. This network evolves in time through a system of ordinary differential equations for the edge weights. We interpret each edge weight as the strength of the relationship between a pair of individuals, with edges increasing in weight if pairs continually listen to each other’s opinions and decreasing if not. We investigate the impact of various edge dynamics at different timescales on the opinion dynamics itself, both analytically and numerically. We will see that the dynamic edge weights can have a significant impact on the opinion formation process since they may result in consensus formation but can also reinforce polarisation. Overall, the proposed modelling approach allows us to quantify and investigate how the network and opinion dynamics influence each other and ultimately allows us to design control methodologies that steer the population to desired states.

ABSTRACT. Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs whose size tends to infinity. We introduce transfer operators, such as the Koopman operator and Perron-Frobenius operator, associated with random walk processes on graphons and illustrate how these operators can be estimated from data and how their eigenvalues and eigenfunctions can be used for detecting clusters, thereby extending conventional spectral clustering methods from graphs to graphons. Furthermore, we show that it is also possible to reconstruct transition probability densities and, if the random walk process is reversible, the graphon itself using only time-series data. The resulting data-driven methods are applied to a variety of synthetic and real-world signals, including daily average temperatures and stock index values.

ABSTRACT. Temporal networks provide a natural framework for modeling complex systems with time-dependent interactions, where understanding the evolution of community structure is a central challenge. While dynamical approaches to community detection in static networks are well established through the spectral analysis of transfer operators, extending these ideas to temporal networks is nontrivial due to the inherent time-dependence of the underlying dynamics. In this work, we develop a general framework for community detection in temporal networks based on multiview Canonical Correlation Analysis (CCA). We show that the proposed formulation can be solved via the spectral analysis of a time-reversible random walk on an augmented space–time network, providing a clear dynamical interpretation of temporal communities as metastable structures of the associated process. Furthermore, we analyze key spectral properties of the resulting transfer operator and identify the interplay between spatial and temporal effects, enabling a principled separation of structural features from artifacts induced by snapshot coupling. Finally, we derive a reduced-order model, which preserves the essential spectral characteristics while significantly improving computational efficiency. In numerical experiments, we demonstrate that the proposed approach effectively detects communities and captures their evolution in temporal networks.

ABSTRACT. We propose a variance reduction method for calculating transport coefficients in molecular dynamics using an importance sampling method via Girsanov's theorem applied to Green--Kubo's formula. We optimize the magnitude of the perturbation applied to the reference dynamics by means of a scalar parameter alpha and propose an asymptotic analysis to fully characterize the long-time behavior in order to evaluate the possible variance reduction. Theoretical results corroborated by numerical results show that this method allows for some reduction in variance, although rather modest in most situations.

ABSTRACT. We will consider the dynamics of dissipative systems with stochastic forcing and focus on mean-square stability to examine uncertainty. First we examine when the stochastic system is mean-square dissipative. Next we will examine the linearised system and state conditions ensuring that perturbations of a linear system with affine noise are bounded. We then relate the mean-square dynamics of the nonlinear and linearised systems. The approach gives a straightforward deterministic method to examine the effects of stochastic forcing on the stability of equilibria of deterministic systems and to obtain bifurcation diagrams that could be included into standard numerical continuation packages. The idea is illustrated numerically on some standard examples from dynamical systems and some examples arising from computational neuroscience.

ABSTRACT. Randomized numerical methods have beneficial properties for integrating low-regularity functions. These advantages can be transferred to the numerical solution of an initial value problem. In this talk, we consider a randomized singly diagonally implicit Runge–Kutta (SDIRK) method based on a randomized trapezoidal rule as the underlying quadrature scheme. The method is constructed such that every realization is an A-stable SDIRK scheme of at least second order. This yields improved stability properties compared to previously studied higher-order randomized Runge–Kutta methods. Moreover, we establish an error bound of order 2.5 in the root mean square sense under low-regularity assumptions. Numerical experiments demonstrate the robustness of the proposed method when applied to nonsmooth problems.

This talk is based on joint work with Marvin Jans (Lund University), Raphael Kruse, and Helmut Podhaisky (both Martin Luther University Halle-Wittenberg).

ABSTRACT. Implicit methods are a natural approach for the integration of stiff problems, to avoid timestep restrictions faced by standard explicit integrators. Explicit stabilized integrator are an alternative to implicit methods which can be particularly efficient in high-dimensional applications with diffusive terms, for example recently for magnetohydrodynamics in the context of solar physics. To obtain the best of both worlds, we introduce a new explicit stabilized implementation of singly diagonally implicit Runge-Kutta (SDIRK) methods. This is achieved by reformulating the implicit Runge-Kutta scheme as the equilibrium of a modified system, that is computed using a partitioned Runge-Kutta-Chebyshev method inspired from optimization techniques.

ABSTRACT. We propose a continuous-time scheme for large-scale optimization that introduces individual, adaptive momentum coefficients regulated by the kinetic energy of each model parameter. This approach automatically adjusts to local landscape curvature to maintain stability without sacrificing convergence speed. We demonstrate that our adaptive friction can be related to cubic damping, a suppression mechanism from structural dynamics. Furthermore, we introduce two specific optimization schemes by augmenting the continuous dynamics of mSGD and Adam with a cubic damping term. Empirically, our methods demonstrate robustness and match or outperform Adam on training ViT, BERT, and GPT2 tasks where mSGD typically struggles. We further provide theoretical results establishing the exponential convergence of the proposed schemes.

ABSTRACT. Ordinary differential equation (ODE) solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity and, implicitly, in stochastic optimization. In this work, we propose and study the stochastic Euler dynamics—a continuous-time Markov process that is equivalent to a linear spline interpolation of a random timestep (forward) Euler method. We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE solution that shall be approximated. We first obtain qualitative insights by studying deterministic Euler dynamics, which we derive through a first-order approximation to the infinitesimal generator of the stochastic Euler dynamics. In the context of linear ODEs, these deterministic Euler dynamics describe the dynamics of the expectation of the stochastic Euler dynamics. Then we show convergence of the stochastic Euler dynamics to the ODE solution by studying the associated infinitesimal generators and by a novel local truncation error analysis. Next, we prove stability by an immediate analysis of the random timestep Euler method and by deriving Foster–Lyapunov criteria for the stochastic Euler dynamics; the latter also yield bounds on the speed of convergence to stationarity. The paper ends with a discussion of second-order stochastic Euler dynamics and a series of numerical experiments that appear to verify our analytical results.

ABSTRACT. Mean-field, ensemble-chain, and adaptive samplers have long been treated as separate approaches to Monte Carlo. We unify them in a two-system framework: an ensemble is split into two subsystems that take turns proposing updates for each other. Each subsystem preconditions its proposals using the empirical covariance of the other, learning the target's stationary covariance on the fly. This cross-system symmetry preserves the target as the invariant distribution at finite particle count and in the mean-field limit. When the two-system dynamics are used as an adaptation phase that feeds a rolling covariance estimate into a downstream HMC sampler, the ensemble size need not scale with dimension, a small number of particles suffices to learn the preconditioner. We derive two-system overdamped and underdamped Langevin samplers that use one BCSS-2 integrator step per Metropolis–Hastings accept/reject, instead of the long trajectories of HMC/NUTS. On synthetic and real posterior benchmarks, including high-dimensional ones, the adaptive MAKLA-BCSS-2 variants remain stable and substantially outperform NUTS in effective samples per gradient and wall-clock throughput.

This is joint work with James Chok, Johnny Lee and Geoff Vasil from the University of Edinburgh.

ABSTRACT. The aim of this presentation is to introduce methods for characterizing the law of measure-valued Markov processes (empirical measures of particle systems and conditional laws of McKean-Vlasov–type equations) through their infinitesimal generators (or Cauchy problems). We will use these results to obtain explicit convergence rates in distribution for the convergence of finite particle system to McKean-Vlasov equations.

ABSTRACT. In this talk, we consider the cubic nonlinear Schrodinger equation (NLS) with a spatially rough potential, a key model for nonlinear Anderson localization. Given its importance in simulations, the previous numerical and the analytical study cover only L^2-potential case. We go beyond the limit in this paper for optimal computations, finding and covering the roughest possible potentials within the well-posedness. Our result comprises three main parts: (1) Sharp PDE theory: We establish new global well-posedness and ill-posedness thresholds in Sobolev and Fourier–Besov-type spaces for the NLS with potentials as singular as the Dirac function. We quantify how the regularity of the solution depends explicitly and optimally on that of the potential. (2) Optimal numerical analysis: Based on the delicate PDE theory, we design a new low-regularity integrator tailored to rough potentials, for which we prove convergence rates with sharp regularity dependence. (3) Numerical verification: Our simulations not only confirm the predicted regularity theory and error bounds on scheme but also demonstrate superior performance over traditional schemes in the rough regime.

ABSTRACT. Strichartz estimates have allowed spectacular progress in the analytical study of nonlinear dispersive equations such as nonlinear Schrodinger equations. We show two consequences regarding error estimates for time splitting method: a global in time error estimate, obtained with Chunmei Su, and low regularity error estimates in the presence of an harmonic potential.

ABSTRACT. Approximating the matrix exponential is central to the numerical simulation of dispersive and highly oscillatory systems. In many applications, computational constraints make the ability to take large time steps more critical than high-order accuracy.

Since Taylor-type analyses, including Lie algebraic approaches, inherently assume small time steps, they are not suitable for constructing methods that remain effective at large step sizes. Polynomial Krylov methods face a related limitation: while capable of high accuracy, their performance is sensitive to the operator norm and is effective primarily for small time steps. These challenges are further exacerbated in the presence of unbounded differential operators or singular potentials such as the Coulomb interaction.

We present three approaches for this setting: analytic methods based on unitary rational best approximations, adaptive strategies for rational Krylov methods, and learned splitting methods. These techniques enable accurate large-step integration while preserving key geometric properties.

ABSTRACT. This paper studies the numerical solution of the semiclassical nonlinear Schr\"odinger equation on the $d$-dimensional torus $\mathbb{T}^d$, with highly oscillatory initial data depending on a small parameter $\varepsilon \in (0,1]$. We first show that a WKB-type approximation attains an $\mathcal{O}(\varepsilon)$ error in the $L^2$ norm for $H^2$ initial data theoretically, although its accuracy deteriorates as $\varepsilon$ increases. To address this limitation, we propose a numerical scheme that (i) applies a Galilean transform to remove the oscillations in the initial data, (ii) establishes sharp space--time estimates for the transformed equation, and (iii) employs a new low-regularity integrator to achieve second-order accuracy under the minimal $H^2$ regularity, which is weaker than the regularity assumptions in the literature. Furthermore, our analysis shows that the CFL-type conditions linking $h$, $\tau$, and $\varepsilon$ --- typically imposed in the semiclassical regime in the literature --- are not required in our scheme to obtain second-order convergence with respect to $\tau$ and $h$, uniformly with respect to $\varepsilon$, under the weaker regularity condition. Numerical experiments support the theoretical results and demonstrate the robustness of the method across a wide range of $\varepsilon$.

ABSTRACT. In machine learning-based molecular force fields, small-scale training data are typically derived from microscopic calculations. Consequently, mitigating modeling errors and enhancing computational precision are prerequisite to the accurate capture of molecular interactions. While the Schrödinger equation represents the most fundamental and precise model, its practical application is frequently hindered by the curse of dimensionality, the singularity of the Coulomb kernel, and the strict requirements of antisymmetric constraints. Consequently, we propose an accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schrödinger equation. The SOG-TNN utilizes a low-rank tensor product representation of the solution to overcome the curse of dimensionality associated with high-dimensional integration. To handle the Coulomb interaction, we introduce an SOG decomposition to approximate the interaction kernel such that it is dimensionally separable, leading to a tensor representation with rapid convergence. We further develop a range-splitting scheme that partitions the Gaussian terms into short-, long-, and mid-range components. They are treated with the asymptotic expansion, the low-rank Chebyshev expansion, and the model reduction with singular-value decomposition, respectively, significantly reducing the number of two-dimensional integrals in computing electron-electron interactions. The SOG decomposition well resolves the computational challenge due to the singularity of the Coulomb interaction, leading to an efficient algorithm for the high-dimensional problem under the TNN framework. Numerical results demonstrate the outstanding performance of the new method, revealing that the SOG-TNN is a promising way for accurately tackling quantum systems, and enables the generation of high-fidelity training data for machine learning potentials.

ABSTRACT. Fast Ewald summation is the most widely used approach for computing long-range Coulomb interactions in molecular dynamics (MD) simulations. While the asymptotic scaling is nearly optimal, its performance on parallel architectures is dominated by the global communication required for the underlying fast Fourier transform (FFT). Here, we develop a novel method, ESP - Ewald summation with prolate spheroidal wave functions (PSWFs) - that, for a fixed precision, sharply reduces the size of this transform by performing the Ewald split via a PSWF. In addition, PSWFs minimize the cost of spreading and interpolation steps that move information between the particles and the underlying uniform grid. We have integrated the ESP method into two widely-used open-source MD packages: LAMMPS and GROMACS. Detailed benchmarks show that this reduces the cost of computing far-field electrostatic interactions by an order of magnitude, leading to better strong scaling with respect to number of cores. The total execution time is reduced by a factor of 2 to 3 when using more than one thousand cores, even after optimally tuning the existing internal parameters in the native codes. We validate the accelerated codes in realistic long-time biological simulations.

ABSTRACT. We discuss some recent progress in the development of random batch sampling and kernel approximation methods for efficient MD simulations of particles under quasi-2D (slab) confinement. Tailored schemes are developed to effectively address the long-range nature of interaction kernels, as well as the anisotropy of confined systems. Numerical examples will be presented to demonstrate the promise of our method as a powerful tool for large-scale simulations of particle systems under confinement.

ABSTRACT. Efficient Monte Carlo (MC) sampling of many-body systems with long-range electrostatics is often limited by the cost of per-move energy-difference evaluation under periodic boundary conditions. In this talk, I introduce the DMK-MC, an accelerated MC method that adapts the dual-space multilevel kernel-splitting (DMK) framework to single-particle Metropolis updates. DMK-MC computes the energy change with an overall O(log N) expected work per trial move for fixed accuracy. Benchmarks on uniform, highly non-uniform, and implicit-solvent electrolyte and colloidal configurations show that DMK-MC consistently outperforms a recent FMM-based O(log N) Monte Carlo method, delivering several-fold speedups at comparable tolerances.

ABSTRACT. In this talk we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of an electro-magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle.

For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge–Kutta method.

We conclude the talk with numerical experiments to explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of a hyperbolic Penning trap.

References

S. Burkhard, B. Dörich, M. Hochbruck, and C. Lasser. “Variational Gaussian approximation for the magnetic Schrödinger equation”. In: Journal of Physics A: Mathematical and Theoretical 57 (2023). https://doi.org/10.1088/1751-8121/ad591e

M. Scheifinger, K. Busch, M. Hochbruck, and C. Lasser. “Time-integration of Gaussian variational approximation for the magnetic Schrödinger equation”. https://arxiv.org/abs/2504.03407

ABSTRACT. The Gross-Pitaevskii equation (GPE) and the magnetic Ginzburg-Landau equation (GLE) plays central role in various models of superfluids and condensed matter physics. A dominating feature is the occurrence of quantized vortices that effectively evolve according to a Hamiltonian system in the limit of point-like vortices. We present a new numerical method that exploits this analytical structure by numerically propagating the vortex-positions and reconstructing the wave-function. We present quantitative error estimates of the so-obtained approximation based on the involved (numerical) parameters. Finally, we will present numerical examples.

ABSTRACT. Hagedorn functions are carefully constructed generalizations of Hermite functions to the setting of many-dimensional squeezed and coupled harmonic systems [1]. I will describe recent developments that allowed application of Hagedorn wavepackets to vibronic spectroscopy of even anharmonic systems. To evaluate time correlation functions needed for computing spectra, we first derived efficient recursive expressions for the overlaps between Hagedorn bases associated with different Gaussians [2]. To succinctly highlight advantages of Hagedorn wavepackets, I will focus on applications to single vibronic level (SVL) fluorescence experiments, in which the electronically excited initial state is also excited in one or several vibrational modes. In displaced, squeezed, and Duschinsky-rotated globally harmonic systems, Hagedorn functions are exact solutions to the time-dependent Schrödinger equation and can be propagated with the same equations of motion as a simple Gaussian wavepacket; emission spectra from arbitrary vibronic levels can be evaluated using a single trajectory. After validating the method by comparing it with exact quantum calculations [3], we applied it to compute SVL spectra of anthracene by performing wavepacket dynamics on a 66-dimensional harmonic potential energy surface constructed from density functional theory calculations [4]. However, real molecules have anharmonic surfaces. To partially describe effects of anharmonicity on spectra, we combined the Hagedorn approach with local harmonic approximation of the potential [5] and with on-the-fly ab initio dynamics, which allowed us to compute SVL fluorescence spectra of difluorocarbene, a floppy molecule with a very anharmonic potential energy surface [6]. Time permitting, I will also briefly mention other improvements and applications to other spectroscopies, such as evaluating Herzberg-Teller and resonance Raman spectra.

[1] G. A. Hagedorn, Ann. Phys. 269, 77 (1998). [2] J. J. L. Vaníček and Zhan Tong Zhang, J. Phys. A: Math. Theor. 58, 085303 (2025). [3] Z. Tong Zhang and J. J. L. Vaníček, J. Chem. Phys. 161, 111101 (2024). [4] Z. Tong Zhang and J. J. L. Vaníček, J. Chem. Theory Comput. 21, 9726 (2025). [5] Z. Tong Zhang, M. Visegrádi, and J. J. L. Vaníček, Mol. Phys., e2577109 (2025). [6] Z. Tong Zhang, M. Visegrádi, and J. J. L. Vaníček, Phys. Rev. A 111, L010801 (2025).

ABSTRACT. Understanding the real-time dynamics of driven-dissipative quantum many-body systems is a central challenge of modern many-body physics, and a prerequisite for harnessing quantum computation and simulation, sensing and dissipative state engineering on platforms such as Rydberg atom arrays. The combined obstacles of strong long-range interactions, two-dimensional geometries and the quadratic cost of working with density matrices place these regimes outside the reach of most existing numerical methods.

In this talk, I will present a tree tensor network (TTN) approach to the Lindblad dynamics of dissipative quantum spin systems on one- and two-dimensional lattices. The hierarchical, loop-free structure of TTNs admits exact and efficient contraction, while connecting any two sites through only a logarithmic number of intermediate tensors, which is particularly advantageous for long-range interactions and 2D layouts where MPS become inefficient and PEPS contractions become prohibitive. Time evolution is performed within the framework of dynamical low-rank approximation, using the Basis-Update and Galerkin (BUG) integrator, which avoids the backward-in-time substep of the projector-splitting/TDVP scheme that destabilises strongly dissipative dynamics, and exposes a natural parallelism on tree topologies. I will benchmark the approach against exact and semiclassical methods, outlining the boundary between the regimes where approximate schemes suffice and those where a tensor-network treatment becomes necessary.

ABSTRACT. The high dimensionality of kinetic equations with stochastic parameters poses major computational challenges for uncertainty quantification (UQ). Traditional Monte Carlo (MC) sampling methods, while widely used, suffer from slow convergence and high variance, which become increasingly severe as the dimensionality of the parameter space grows. To accelerate MC sampling, we adopt a multiscale control variates strategy that leverages low-fidelity solutions from simplified kinetic models to reduce variance. To further improve sampling efficiency and preserve the underlying physics, we introduce surrogate models based on structure and asymptotic preserving neural networks. These deep neural networks are specifically designed to satisfy key physical properties, including positivity, conservation laws, entropy dissipation, and asymptotic limits.The proposed methodology enables efficient large-scale prediction in kinetic UQ. Numerical results demonstrate improved accuracy and computational efficiency compared to standard MC techniques.

ABSTRACT. The study of plasma models is receiving a great deal of attention from the scientific community because of its potential applications to nuclear fusion reactors in the search for new energy sources. At the kinetic scale, the time evolution of the distribution functions of charged particles in plasmas is described by the Landau-Fokker-Planck equation. Among the various numerical approaches to solving collisional kinetic equations, particle methods are widely used, especially for high-dimensional simulations, where the computational cost may represent a bottleneck. Indeed, particle methods scale well with the dimensionality and they are able to capture the multiscale structures of these equations, from rarefied interacting regimes to the hydrodynamic limit. In this talk, we will explore and compare different particle-based techniques to solve the Landau equation, from deterministic particle method based on a gradient flow reformulation of the collisional operator, to Monte Carlo schemes.

ABSTRACT. In this talk, I will present a new tensor train approach for computing Boltzmann collision operator. We lift the Boltzmann equation into higher dimensional ambient space, and compute the lifted equation by tensor train cross approximation. We perform projection correction for mass momentum energy conservation. We demonstrate the effectiveness of the computational methods by benchmark tests.

ABSTRACT. Structure-preserving particle methods have recently been proposed for a class of aggregation-diffusion equations and the Landau equation. Both of such models can be viewed as a class of nonlinear continuity equation with a velocity field depending on the variational derivative of some energy functional. While they can be semi-discretized using particle methods to satisfy energy dissipation and preservation of conserved quantities with suitable regularization, conventional time integrators applied to such semi-discretizations do not preserve these properties in general.

In this talk, we introduce a notion of compatibility condition for the regularized energy functional which will enable the variational derivatives to be expressed in terms of the gradient of some particle energy functional. This approach will allow us to make use of discrete gradient integrators and to show the resulting full-discretization preserves energy dissipation and conserved quantities simultaneously. We demonstrate the dissipative and conservative properties of our method on various numerical examples. In addition, we showcase the decay of Fisher information and entropy dissipation rate in the case of the Landau equation with the Coulomb kernel.

If time permits, we will also discuss the application of Anderson Acceleration to efficiently solve these implicit structure-preserving schemes.

ABSTRACT. Deep learning has made significant progress in data science and natural science. Some studies have linked deep neural networks to dynamical systems, but the network structure is restricted to residual networks. It is known that residual networks can be regarded as numerical discretizations of dynamical systems. In this talk, we consider traditional network structures and prove that vanilla feedforward networks can also be used for the numerical discretization of dynamical systems, where the width of the network is equal to the input and output dimensions. The proof is based on the properties of the leaky-ReLU function and the numerical technique of the splitting method for solving differential equations. The results could provide a new perspective for understanding the approximation properties of feedforward neural networks. In particular, the minimum width of neural networks and the minimal control family of dynamical systems for universal approximation can be derived. In addition, the relationship between mapping compositions and regular languages can be established.

ABSTRACT. We study the problem of approximating a given diffeomorphism and its associated pushforward probability measure using the flow of a Neural ODE with a piecewise-constant velocity field. This setup models conditional sampling tasks where one seeks both an inverse map and a valid generative transform. The core contribution is an explicit constructive method: decomposing the target map into compressible and incompressible components and realizing each via simple flow elements. The compressible part is captured exactly through the gradient of a convex potential, while the incompressible part is realized via shear flows and permutations. Under stronger regularity assumptions on the target map (e.g., for the Knöthe–Rosenblatt rearrangement with regular input and output measures), we provide a probabilistic construction that avoids the exponential scaling (“curse of dimensionality”) in the number of parameters in the neural network.

This is a joint work with Borjan Geshkovski

ABSTRACT. We investigate the expressive power of deep residual neural networks idealized as continuous dynamical systems through control theory. Specifically, we consider two properties that arise from supervised learning, namely universal interpolation- the ability to match arbitrary input and target training samples- and the closely related notion of universal approximation- the ability to approximate input-target functional relationships via flow maps. Under the assumption of affine invariance of the control family, we give a characterisation of universal interpolation, showing that it holds for essentially any architecture with non-linearity. Furthermore, we elucidate the relationship between universal interpolation and universal approximation in the context of general control systems, showing that the two properties cannot be deduced from each other. At the same time, we identify conditions on the control family and the target function that ensures the equivalence of the two notions.

ABSTRACT. Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. While recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying Filippov-type piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires not only recovering the governing equations but also detecting the switching manifolds that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a workflow for discovering such systems by first estimating switching manifolds from data and then learning smooth dynamics within each region using neural networks combined with physics-informed regularisation. The approach is tested on low-dimensional benchmark problems, including the dry-friction oscillator, where it can recover both the switching structure and the underlying dynamics with good accuracy.

ABSTRACT. The computational burden of training and inferencing large scale transformer-based neural networks, e.g. large language models (LLMS) or vision transformers (ViTs), is significant. This motivates the search for memory efficient representations of the weights of these networks, and robust ways to train them.

In particular we consider the attention layer of transformer models, where recent advances as "multi-head-latent-attention" (MLA) demonstrated that low-rank factorizations are viable for state-of-the-art LLMs. We generalize this approach to canonical tensor factorizations, demonstrate an efficient GPU implementation of a low-rank aware flash attention kernel, and discuss the advantages and pitfalls of training this representation with dynamical low-rank approximation methods.

ABSTRACT. High-dimensional problems have system sizes that increase exponentially with increasing resolution, such that resolutions in practical applications give rise to systems with exorbitant storage costs. An example of such a system is that of a collisional plasma described by the kinetic, non-linear Vlasov-Poisson-Fokker-Planck (VPFP) equation, where the phase space consists of position and velocity. The highest computational cost in solving this equation arises from evaluating the collision term in velocity space. We solve the three-dimensional VPFP equation in the Rosenbluth-MacDonald-Judd (RMJ) form, in particular using dynamical low-rank approximation (DLRA), which has been successfully employed in tackling related kinetic equations. In DLRA, the starting high-dimensional problem is projected into several lower-dimensional problems, the size of each of which is subject to the truncated rank. These lower-dimensional problems can be solved using standard methods. Several aspects are addressed to ensure the scalability of our method. For example, the RMJ form of the collision term admits efficient evaluation via solving auxiliary Poisson equations in velocity space. Moreover, local basis functions are employed in velocity space, which allows for sparse matrix calculations. We conclude with numerical experiments for classical benchmarks in plasma physics.

ABSTRACT. Deployment of neural networks on resource-constrained devices demands models that are both compact and robust to adversarial inputs. However, compression and adversarial robustness often conflict. In this work, we introduce a dynamical low-rank training scheme enhanced with a novel spectral regularizer that controls the condition number of the low-rank core in each layer. This approach mitigates the sensitivity of compressed models to adversarial perturbations without sacrificing accuracy on clean data. The method is model- and data-agnostic, computationally efficient, and supports rank adaptivity to automatically compress the network at hand. Extensive experiments across standard architectures, datasets, and adversarial attacks show the regularized networks can achieve over 94% compression while recovering or improving adversarial accuracy relative to uncompressed baselines.

ABSTRACT. Dynamical low-rank approximation (DLRA) compresses large matrix ODEs by evolving the dynamics on the manifold of rank-r matrices. Classical integrators (projector-splitting, BUG) rely on Householder QR of tall factors at every step, which becomes a synchronization bottleneck on parallel and GPU hardware. Randomized numerical linear algebra offers a single-sync, BLAS-3 alternative via randomized Gram-Schmidt - but where to plug it into DLRA turns out to matter. We compare two natural candidates: an oblique variant that sketches the Galerkin condition, and an orthogonal variant that sketches only the basis parameterization. We show that only the orthogonal one preserves the classical DLRA dynamics unconditionally; the oblique one silently perturbs them unless the problem is low-rank compatible. Numerical experiments on Allen-Cahn, Fokker-Planck, and Vlasov-Poisson illustrate the contrast, with a clean energy drift on the non-compatible case.

ABSTRACT. This talk presents weighted finite difference methods for numerically solving a semiclassically scaled cubic nonlinear Schrödinger equation, starting from initial data that are a sum of modulated highly oscillatory exponentials with wave numbers inversely proportional to the semiclassical small parameter. The interaction of waves through the nonlinearity is often referred to as wave mixing. It is of significant mathematical and physical interest, for example in fiber optics.

The proposed numerical methods do not need to resolve high-frequency oscillations in both space and time by prohibitively fine grids as would be required by standard finite difference methods. The approach taken here modifies traditional finite difference methods by appropriate exponential weights. Specifically, we propose weighted leapfrog and weighted Crank-Nicolson methods, both of which achieve second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by the small semiclassical parameter. They are as simple to implement as classical finite difference methods.

The derivation and analysis of the numerical methods relies on modulated Fourier expansions in a Wiener algebra functional setting. Numerical experiments illustrate the theoretical results.

ABSTRACT. We consider a nonlinear Klein–Gordon equation in the nonrelativistic limit with modulated highly oscillatory exponential initial data. In this regime of a small scaling parameter, the solution exhibits rapid oscillations in both space and time, which pose challenges for numerical approximation. We propose explicit and implicit exponentially weighted finite difference methods. The explicit weighted leapfrog scheme satisfies a CFL-type stability condition, while the implicit weighted Crank–Nicolson scheme is unconditionally stable. Both methods achieve second-order accuracy without time step or mesh size restrictions imposed by the scaling parameter. They are uniformly convergent for arbitrarily small to moderately bounded values of the scaling parameter.

ABSTRACT. High-frequency wave propagation in dispersive media is described by semilinear Friedrichs systems with solutions that exhibit rapid oscillations in both space and time. The oscillations result from a small parameter ε whose inverse appears explicitly in the governing equations as well as in the initial data. This oscillatory behaviour of the solutions makes the numerical computation of acceptable approximations extremely expensive. In order to cope with these challenges, we derive a modulated Fourier expansion that achieves an error of order ε^2. The approximation is formulated in terms of several coefficient functions combined with prescribed oscillatory factors. The coefficients are solutions of a system of Schrödinger-type PDEs coupled with an algebraic equation. The solutions of these equations neither oscillate in space nor in time, which is particularly advantageous for numerical approximation.

ABSTRACT. In this talk, we consider the semiclassical Schrödinger equation with uncertain parameters and study multi-fidelity methods to numerically solve it. We employ the time-splitting Fourier pseudospectral method as the high-fidelity solver, and explore different low-fidelity models. The error estimate for the bi-fidelity method and empirical error bounds are shown. Several numerical experiments are conducted to show the accuracy and efficiency of our proposed schemes.

ABSTRACT. Uncertainty quantification (UQ) is essential for reliable scientific machine learning, especially under sparse and noisy observations and in the presence of model misspecification. This talk presents a coherent progression of methods toward trustworthy learning of physical systems. We first develop an information-bottleneck UQ approach for operator learning, using a confidence-aware encoder and a Gaussian decoder to produce calibrated predictive means and variances without costly posterior sampling. We then extend the confidence-aware latent representation to physics-informed learning for PDEs under sparse and noisy data via a latent variable model coupled with Gaussian processes, enabling uncertainty-aware forward and inverse modeling with soft physics constraints. Finally, we introduce a latent-space model-correction strategy with dual decoders for the solution and a discrepancy term, providing joint UQ of both within a single training pipeline. Numerical experiments demonstrate improved robustness and efficiency across representative operator-learning and PDE benchmarks.

ABSTRACT. We discuss some recent work on constructing stable and interpretable macroscopic dynamics from trajectory data using deep learning. We adopt a modelling approach: instead of generic neural networks as functional approximators, we use a model-based ansatz for the dynamics following a suitable generalisation of the classical Onsager principle for non-equilibrium systems. This allows the construction of macroscopic dynamics that are physically motivated and can be readily used for subsequent analysis and control. We discuss applications in the analysis of polymer stretching in elongational flow. Moreover, we will also discuss some algorithmic challenges associated with learning (macroscopic) dynamics for scientific applications.

ABSTRACT. In this talk, we will present our recent work on Fast Equivariant Imaging (FEI), a novel unsupervised learning framework to rapidly and efficiently train deep imaging networks without ground-truth data. From the perspective of reformulating the Equivariant Imaging based optimization problem via the method of Lagrange multipliers and utilizing plug-and-play denoisers, this novel unsupervised scheme shows superior efficiency and performance compared to the vanilla Equivariant Imaging paradigm. In particular, our FEI schemes achieve an order-of-magnitude (10x) acceleration over standard EI on training U-Net for X-ray CT reconstruction and image inpainting, with improved generalization performance. In addition, the proposed scheme enables efficient test-time adaptation of a pretrained model to individual samples to secure further performance improvements. Extensive experiments show that the proposed approach provides a noticeable efficiency and performance gain over existing unsupervised methods and model adaptation techniques.

ABSTRACT. We develop a quantum algorithm for solving linear algebraic equations Ax = b from the perspective of Schrödingerization-form problems, which are characterized by a system of linear convection equations in one higher dimension. This approach realizes the Schrödingerization of quantum linear system problems. Our method can be viewed as an ODE-based quantum algorithm for linear algebraic equations, as the solution to the convection equations corresponds to the steady-state solution of an associated system of linear ODEs. We provide a detailed implementation and explicit error analysis. Additionally, we incorporate a block preconditioning technique to achieve nearly linear scaling in the condition number, leading to near-optimal query complexity.

ABSTRACT. In this work, we propose a numerical homogenization method for scalar linear partial differential equations with rough coefficients, that integrates classical coarse-scale solvers with quantum subroutines for fine-scale corrections. Inspired by the Localized Orthogonal Decomposition [A. Målqvist and D. Peterseim, “Numerical Homogenization by Localized Orthogonal Decomposition”, SIAM (2020)], we employ quantum local problem solvers to capture fine-scale features efficiently. Crucially, the approach does not rely on the periodicity of the problem, and the integration of the quantum computation within a coarse model requires only selected measurements of the quantum representative volume elements, overcoming the information bottleneck of quantum interfaces that could eliminate the speed-up. We demonstrate that the local quantum solver, built on [M. Deiml and D. Peterseim, “Quantum realization of the finite element method”, Mathematics of Computation (2025)], achieves solutions with the required level of accuracy, while the number of operations scales only logarithmically with the fine-scale resolution, determined by the smallest length scale encoded in the diffusion coefficient. The potential of the approach is illustrated through two-dimensional test cases, with periodic and non-periodic highly oscillatory coefficients, using a classical simulation of the local quantum solver.

ABSTRACT. We develop a pivot-shifted Carleman linearization framework for quantum algorithms solving quadratic nonlinear ordinary differential equations. The key idea is to shift the dynamics by a pivot state and perform Carleman lifting in the shifted coordinates, combined with a Lyapunov transform and rescaling. This approach substantially enlarges the class of systems that can be simulated efficiently: for systems that become stable after pivot shifting, we establish long-time convergence of the truncated Carleman embedding without requiring a lower bound on the initial condition in the shifted coordinates. Under explicit spectral and norm conditions on the shifted system, we show that the truncation order scales only logarithmically with the simulation time and target precision, and we derive end-to-end quantum query complexity bounds for preparing a state proportional to the final solution. For more general systems that remain unstable after pivot shifting, we prove short-time convergence guarantees while still removing the restrictive dependence on the initial value present in earlier analyses. Numerical experiments on the logistic equation and the Lotka–Volterra system demonstrate that an appropriate pivot choice can dramatically improve stability and accuracy, yielding exponential error decay with truncation order in favorable regimes. These results show that pivot switching provides a practical and theoretically justified route for extending Carleman-linearization-based quantum algorithms to a broader class of nonlinear dynamical systems.

ABSTRACT. We propose hybrid quantum--classical algorithms for complex-valued nonlinear partial differential equations in strongly nonlinear vortex regimes, where the dynamics is governed by vortex cores, phase singularities, and nonlinear vortex interactions. Typical examples include the complex-valued nonlinear Schrödinger equation, nonlinear heat equations, and nonlinear wave equations with Ginzburg--Landau-type nonlinearities. In these regimes, the leading-order asymptotic dynamics decomposes into a low-dimensional vortex motion law coupled to a high-dimensional linear boundary-value problem for a smooth correction field. Our algorithms exploit this asymptotic structure: the vortex dynamics is advanced classically, while the resulting linear elliptic problem is treated by quantum algorithms. For the two-dimensional nonlinear Schrödinger equation, we combine BPX preconditioning with Schrödingerization to estimate physically relevant observables in the small-output regime. This yields, already in two dimensions, an exponential improvement in the dependence on the spatial problem size, while the dependence on the target accuracy remains essentially linear up to polylogarithmic factors. We further show that the same principle extends to dissipative Ginzburg--Landau vortex dynamics and to vortex filaments in three-dimensional superconductivity. Numerical results support the validity of the nonlinear-to-linear reduction and the effectiveness of the proposed framework.

ABSTRACT. In this talk, we develop a discrete mechanics framework for dissipative dynamics based on the Herglotz variational principle and contact mechanics. We prove that for general contact systems a classical Runge–Kutta method cannot generate a contact map.

Within this framework, we construct Herglotz variational integrators and achieve numerical discretization with high-order accuracy through variational composition techniques. A discrete contact Noether theorem is proved, establishing a connection between the contact Noether quantity and the dissipation law of the system. A generating-function approach is also developed.

In numerical experiments, we illustrate the long-term simulation capability of the newly derived Herglotz variational integrators for an ellipsoid geodesic system. We also demonstrate their connection to optimization approaches, with implementation carried out through a Bregman-type model.

ABSTRACT. In this talk, a new numerical method to approximate the solution of the Vlasov-Maxwell equations is presented. The method uses a phase space discretization and its main properties are: energy and charge conservation thanks to a semi-implicit treatment of the Maxwell equations, but allowing for an explicit and efficient update of the unknown. One of the main ingredients lies in the introduction of an auxiliary scalar variable inspired from the Scalar Auxiliary Variable (SAV) approach together with a suitable splitting which enables the use of a semi-Lagrangian method.

ABSTRACT. Quasicrystals are materials in which the atoms are ordered (in the sense of having a well-defined diffraction pattern) but not periodic, like in a Penrose tiling. The behaviour of electrons in a quasicrystal is governed by the Schrödinger equation with a quasiperiodic potential. As a model, we compute the spectrum of the one-dimensional Schrödinger operator −(d/dx)² + V(x) with a quasiperiodic potential like V(x) = cos(x) + cos(τx) where τ is an irrational number.

There are many methods for computing the spectrum of Schrödinger operators, but they assume that either the potential is either periodic or that the eigenstates are localized. Thus, one approach is to use a periodic approximant for the potential and then (for instance) expand in a Fourier series. We will compare this with another approach, which is to expand in a Fourier–Bohr series using a basis of quasiperiodic functions. Both methods show that the spectrum has an intricate Cantor-like structure as suggested by results from functional analysis. The relative performance of the two methods depends on which part of the spectrum is considered.

ABSTRACT. Particle methods are a powerful numerical method for solving partial differential equations, alongside finite element and finite difference methods. However, because the motion of virtual particles is equivalent to a spatial mesh evolving over time, conventional structure-preserving methods that discretize partial integrals using a fixed mesh cannot be applied. In response to this, we consider discretizing and preserving the variational structure, including the changing spatial mesh, by using a Voronoi mesh with particle positions as generating points and calculating variations for small changes in the Voronoi mesh. We plan to present examples of applications of the Allen-Cahn and Cahn-Hilliard equations.

ABSTRACT. The use of artificial neural networks is now becoming widespread across a wide variety of tasks. In this context of very rapid development, issues related to the storage and computational performance of these models emerge, since neural networks are sometimes very deep and comprise up to trillions of parameters. For all these reasons, the use of reduced precision is increasingly being considered, although, until now, its accuracy and robustness had been approached mostly from a practical standpoint or verified by software.

The aim of this work is to provide formal tools to better understand, explain, and predict the accuracy and stability of neural networks when using floating-point arithmetic. To this end, we apply a rounding error analysis based on existing tools in numerical linear algebra to obtain both forward and backward error bounds. This includes both deterministic worst-case bounds as well as probabilistic bounds that are sharper on average. These bounds both ensure the proper functioning of neural networks once trained and provide recommendations on architectures and training methods to enhance the robustness of neural networks when using floating-point arithmetic.

ABSTRACT. Low-rank classifiers are ubiquitous in modern AI. For example, Large Language Models (LLMs) have vocabularies of hundreds of thousands of tokens, yet their feature activations have far fewer dimensions, making their output layer low-rank. In this talk I will demonstrate a vulnerability of such low-rank classifiers: they can have unargmaxable outputs, i.e. outputs that are impossible to predict irrespective of the input. I will then introduce linear programmes to detect unargmaxable outputs in LLMs and text classification models.

ABSTRACT. Neural networks are known to be vulnerable to adversarial attacks - an imperceptible perturbation to an image can cause a change in classification. Standard attack algorithms use explicit or approximate partial derivative information with respect to the input data. Here, we explore the idea of using a less expensive, universal affine surrogate. In practice, affine attack models tend to be approximately low rank. Truncating to a precisely low rank transformation does not degrade the performance of the model.

ABSTRACT. Models that incorporate jumps are essential for capturing abrupt price movements observed in financial markets. While jump–diffusion dynamics provide a more realistic description of asset behaviour than pure diffusions, they add computational complexity to accurately price options. This difficulty is especially pronounced for American options, which dominate practical trading but have received comparatively less attention in the academic literature than their European counterparts.

Recent advances in machine learning offer new opportunities for addressing these challenges. In particular, neural methods for solving backward stochastic differential equations provide a simulation-based alternative to classical PDE and PIDE techniques, which struggle to scale to high-dimension from the curse of dimensionality. In this talk, we will present a deep-learning framework for pricing American options under multi-asset jump–diffusion models. We derive a discrete-time Backward Stochastic Differential Equation (BSDE) that incorporates both diffusion and jump components and integrate it into the existing diffusion-only deep neural network framework. The resulting method achieves accurate and stable performance in dimensions up to d = 100, demonstrating that modern deep-learning tools can make high-dimensional American option pricing feasible in realistic jump–diffusion settings.

ABSTRACT. We study a continuous time stochastic optimal control problem under partial observations that are available only at discrete time instants. This hybrid setting, with continuous dynamics and intermittent noisy measurements, arises in applications ranging from robotic exploration and target tracking to epidemic control. We formulate the problem on the space of beliefs (information states), treating the controller's posterior distribution of the state as the state variable for decision making. On this belief space we derive a Pontryagin maximum principle that provides necessary conditions for optimality. The analysis carefully tracks both the continuous evolution of the state between observation times and the Bayesian jump updates of the belief at observation instants. A key insight is a relationship between the adjoint process in our maximum principle and the gradient of the value functional on the belief space, which links the optimality conditions to the dynamic programming approach on the space of probability measures. The resulting optimality system has a prediction and update structure that is closely related to the unnormalised Zakai equation and the normalised Kushner-Stratonovich equation in nonlinear filtering. Building on this analysis, we design a particle based numerical scheme to approximate the coupled forward (filter) and backward (adjoint) system. The scheme uses particle filtering to represent the evolving belief and regression techniques to approximate the adjoint, which yields a practical algorithm for computing near optimal controls under partial information. The effectiveness of the approach is illustrated on both linear and nonlinear examples and highlights in particular the benefits of actively controlling the observation process.

ABSTRACT. This paper mainly studies global $\mu$-consensus in the mean square problem of the nonlinear multiagent systems (MASs) with unbounded time-varying delays and stochastic delayed impulses. Different from previous works, stochastic delayed impulsive effects are considered in general nonlinear delayed MASs to simulate the sudden changes that the systems encounter during operation. In addition, a novel event-triggered mechanism is designed, where it is only necessary to monitor the state of the MASs at impulsive instants. In this way, Zeno phenomenon can be effectively avoided. Moreover, the frequency of information exchange among agents and controller updates can be reduced, leading to a reduction in control costs. Then, based on the designed event-triggered control strategy and some techniques of stochastic analysis, the global consensus in the mean square criteria for the nonlinear delayed MASs under stochastic delayed impulses are proposed. Finally, examples are presented to illustrate the validity of the results obtained.

ABSTRACT. We consider the probabilistic interpretation of the Kinetic Fokker-Planck equation in a bounded domain in position, with reflecting boundary conditions. We construct the reflected Kinetic Langevin process associated and we study its long-time behavior depending on the boundary conditions thanks to probabilistic tools (Harris theory). We also focus on the cases where the invariant measure is the Gibbs-measure for example the Langevin process with specular or Maxwellian reflections. This work is a first step towards the analysis of the sampling of a Gibbs measure in a bounded domain for a kinetic Langevin dynamics which would be the discretization of our reflected process.

ABSTRACT. Probabilistic models of shape variation are increasingly popular in computational anatomy and geometric learning, but their practical use is limited by the computational cost and numerical challenges of fitting uncertainty-aware, topology-preserving models at high resolution. We consider statistical shape models in which shapes are characterised as diffeomorphic deformations of a template driven by time-dependent random velocity fields. In high dimensions, posterior inference for such models is computationally demanding, and naive multiresolution strategies can lose accuracy when coarse and fine dynamics are weakly coupled.

We propose a coarse-to-fine acceleration framework that constructs deformation-aware coarsening operators together with a lifting procedure that embeds coarse model and parameters into a fine subspace. By enforcing shared randomness and coupling of distributions across levels, the method generates strongly correlated coarse and fine velocity fields, and hence tightly coupled deformation trajectories. The lifted model also provides an informed initialisation for fine-level optimisation, reducing the number of expensive fine iterations while preserving consistency of the random dynamics across resolutions. We support the method with theoretical analysis for the differences in associated quantities of interest, and numerical experiments show at least a twofold reduction in wall-clock time relative to standard variational fitting, without loss of deformation fidelity.

ABSTRACT. Second derivative hybrid block methods are presented for the treatment of stiffness in Ordinary Differential Equations. The parameters of the method are obtained by employing Taylor's Series expansion and method of undetermined coefficients. Stability analysis is carried out on the developed methods and the methods are A-stable for order at least p≤15. Numerical experiments shows that these methods enrich tool kits for approximating stiff initial value problems.

ABSTRACT. The stochasticity of cellular processes comes from both external environmental factors and internal molecular fluctuations, influencing gene expression and regulation. These are the primary factors determining cellular function and fate. Understanding how these interactions evolve over time is important in order to identify the time varying mechanisms that govern cellular state dynamics.

We formulate the problem statement as a multiscale inference model that infers gene expression at the micro level and cellular trajectories at the macro level. The Ornstein-Uhlenbeck (OU) process governs the stochastic dynamics of mRNA concentrations for a given gene in a GRN. At the same time, the underlying Fokker-Planck (FP) law determines the evolution of the population level cell state distribution. We then use a combined loss function that couples the FP evolution with the underlying OU process via a Wasserstein gradient flow formulation, with an additional sliced Wasserstein term enforcing consistency between the inferred posterior distribution and the empirical distribution of the observed population.

Furthermore, in a first-principles approach we can explicitly account for uncertainty through the posterior distribution. This enables downstream biological analysis to make confident predictions and identify regulatory factors that may not be distinguishable due to noise or incomplete observations.

ABSTRACT. A structure-preserving and accurate simulation of energy networks based on hierarchical port-Hamiltonian formulations has been of profound interest in recent research. In this work, we consider the construction of a model hierarchy, which arises through different approaches. These include physical simplification of the model by neglecting specific effects, algebraic reduction by model order reduction (MOR), and computational refinement through varying mesh resolutions in finite element discretisations. The port-Hamiltonian formulation provides a unifying framework that enables consistent interconnection of models across different hierarchical levels while preserving fundamental properties, such as energy dissipation and mass conservation. In this context, we consider nonlinear flows on network graphs and the improvement in efficiency brought by the model hierarchy proposed.

ABSTRACT. Learning under non-smooth objectives remains a fundamental challenge in machine learning, as abrupt changes in conditioning variables can induce highly non-smooth loss landscapes and destabilize optimization. This difficulty is particularly pronounced in non-autonomous dynamical systems driven by discontinuous inputs, where widely used optimization methods, including recent neural smoothing approaches, exhibit unreliable convergence or strong hyperparameter sensitivity. To address this issue, we propose Deep Predictor–Corrector Networks (DePCoN), a multi-scale learning framework that stabilizes optimization by learning scale-consistent parameter update rules across a hierarchy of smoothed inputs. Rather than treating smoothing as a fixed preprocessing choice, DePCoN integrates smoothing into the learning dynamics itself through a learned predictor–corrector mechanism. Across biological and ecological benchmarks with discontinuous inputs, DePCoN consistently achieves more robust and faster convergence than existing methods while substantially reducing sensitivity to hyperparameter choices. Beyond dynamical systems, our approach provides a general learning principle for stabilizing optimization under non-smooth objectives.

ABSTRACT. We introduce a tamed Milstein scheme for stochastic differential equations with Markovian switching, specifically tailored for cases where both the drift and diffusion coefficients exhibit super-linear growth. We establish the moment stability of the proposed scheme and prove that it achieves a strong convergence rate of 1.0 in the L^p-sense, even when the coefficients are non-globally Lipschitz and grow super-linearly.

ABSTRACT. Sampling from Gibbs measure in high dimensions is a challenging task, especially in the case of multimodals and stiff potentials. Indeed, the natural approach of using Langevin dynamics can converge very slowly to equilibrium. In this work, we consider a Stratonovich perturbation to modify the dynamics and improve the convergence, while preserving the reversibility of the dynamics. We construct an optimal choice of perturbation to improve the convergence rate, which we prove in the two dimensional quadratic case. Numerical comparison with established non-reversible samplers and the overdamped Langevin dynamics shows the efficiency of the approach beyond the quadratic case.

ABSTRACT. The Dean-Kawasaki equation is a fundamental stochastic PDE from the theory of fluctuating hydrodynamics and has been proposed as a model for density fluctuations in large, finite-sized systems of diffusing particles. It is a highly singular SPDE that has been shown to only admit trivial martingale solutions which complicates mathematical approaches to its rigorous justification. In the recent work of Cornalba et al., it was shown that finite difference discretisations - a natural form of regularisation - accurately describe density fluctuations of diffusing particles in suitably weak metrics. In this contribution we significantly widen the applicability of numerical discretisations of the Dean-Kawasaki equation by considering a spatial discontinuous Galerkin (dG) approximation. Our method overcomes the restrictive grid requirements of finite-difference schemes whilst also being locally conservative and amenable to parallelisation. Simulating Dean-Kawasaki type equations is becoming increasingly important in applied sciences for studying fluctuation-driven behaviour in physical systems and this research lays the groundwork for future applications of dG-FEMs to more complex models such as weakly-interacting systems on irregular domains with mixed boundary conditions.

ABSTRACT. The evaluation of non-local operators on unbounded domains, particularly the Hilbert transform, is fundamental to simulating the evolution of dispersive PDEs such as the Benjamin-Ono and complex water-wave equations. While exponentially convergent quadrature exists for functions strictly analytic within a strip containing the real line, computing the Hilbert transform for functions exhibiting non-integer algebraic decay using only limited uniform grid samples remains a formidable numerical challenge.

This work proposes a novel hybrid framework that decouples non-local asymptotic tails from local high-frequency dynamics. Utilizing the Matrix Pencil method, we extract the fractional algebraic decay orders from the domain boundaries. The original discrete signal is then rigorously decomposed into a rapidly decaying residual vector and a set of asymptotic basis functions. For the latter, the extracted exact functional forms allow for ultra-high-precision semi-analytic integration. For the residual, we apply spatial extrapolation followed by a Finite Impulse Response (FIR) discrete Hilbert filter.

By recombining these projections, our method demonstrates rapid error convergence as the sampling window extends and grid spacing decreases, with the global numerical error strictly scaling to the theoretical minimax bound of $O(\epsilon_{mach}^{2/3})$. Finally, we address the ongoing challenges posed by extreme spatial spectral variations in highly nonlinear water waves, outlining a roadmap for embedding this high-precision operator into fully discrete PDE solvers.

ABSTRACT. In the nonrelativistic limit regime, nonlinear Dirac equations involve a small parameter ε>0 which induces rapid temporal oscillations with frequency proportional to ε^(−2). Efficient time integrators are challenging to construct, since their accuracy has to be independent of or not significantly affected by ε. We present three novel methods developed specifically for the Dirac equation.

For the first one, we consider an alternative system which allows for analytical approximations of solutions. This comes with an a-priori error of O(ε^2), but the improved regularity given by this new system enables us to derive a second-order accurate two-step method for it.

To construct the second scheme, we start with the common approach of iterating Duhamel’s formula and then approximating slowly varying parts. This allows constructing a second-order method that is uniformly accurate in ε, but also very complicated due to the nonlinear nature of the problem. Therefore, we propose a simplified version which exploits cancelation effects in the error accumulation. We prove that for non-resonant step sizes, there is no loss of accuracy compared to the full method.

Finally, we suggest a splitting method of a novel type. Unlike in classical splitting schemes, the dominant operator in the PDE is included in every subproblem - and, in some sense, rewound in between them. As a result, our method is significantly less affected by fast oscillations. Since the subproblems are not trivial to solve, we additionally derive efficient integrators for them.

The individual advantages of each method are illustrated in several error plots.

ABSTRACT. We propose a neural differential system with Lie group embedding for compact geometric learning, in which the network state evolves in the Lie algebra while the coupling is induced by group-valued parameters through the adjoint action representation. This avoids the inconsistency of applying ordinary Euclidean weighted summation directly on nonlinear manifolds and gives each interaction a clear geometric meaning as a change of reference frame for infinitesimal motions. The resulting model preserves the symmetry and intrinsic structure of the embedded compact Lie group while retaining a unified continuous-time ODE form. We formulate the framework for matrix Lie groups and their Lie algebras, derive the associated neural evolution equations, and clarify how adjoint-action coupling provides structural closure of the dynamics without relying on Euclidean projections. The proposed perspective offers a mathematically consistent extension of classical neural ODE to non-Euclidean geometric settings, and suggests a principled bridge between Lie theory, neural ODEs, and geometric representation learning. This work is motivated by learning problems with rotationally constrained or symmetry-aware latent variables and aims to provide a structure-preserving foundation for neural computation upon structured geometry.

ABSTRACT. Using an L1 scheme to approximate fractional integrals and derivatives, alongside finite element methods on a piecewise linear grid, approximate the solution to stochastic subdiffusion problems including a laplacian term.

A number of theoretical results are proven alongside the temporal rates of convergences, which are validated with numerical simulations, which are done in MATLAB.

ABSTRACT. We present a framework for solving partial differential equations using physics-informed neural networks (PINNs) defined in Chebyshev coefficient space. By representing solutions spectrally, we enable stable differentiation through recurrence-based Chebyshev derivatives and incorporate boundary conditions with minimal pointwise evaluation. To address slow and unstable convergence in PINNs, we investigate preconditioning strategies based on geometric optimisation with a focus on Natural Gradient Descent. These methods account for the mismatch between parameter-space and function-space geometry, improving optimisation efficiency and training stability. Our results demonstrate that combining spectral representations with geometry-aware optimisation provides a robust approach for learning PDE solutions, and offers a foundation for future operator-learning extensions.

ABSTRACT. Bayesian neural networks (BNNs) require scalable sampling algorithms to approximate posterior distributions over parameters. Existing stochastic gradient Markov Chain Monte Carlo (SGMCMC) methods are highly sensitive to the choice of stepsize and adaptive variants such as pSGLD typically fail to sample the correct invariant measure without addition of a costly divergence correction term. In this work, we build on the recently proposed ‘SamAdams’ framework for timestep adaptation (Leimkuhler, Lohmann, and Whalley 2025), introducing an adaptive scheme: SA-SGLD, which employs time rescaling to modulate the stepsize according to a monitored quantity (typically the local gradient norm). SA-SGLD can automatically shrink stepsizes in regions of high curvature and expand them in flatter regions, improving both stability and mixing without introducing bias. We show that our method can achieve more accurate posterior sampling than SGLD on high-curvature 2D toy examples and in image classification with BNNs using sharp priors.

ABSTRACT. We propose the Long-Short-Range Neural Network (LSR-Net), an extensible operator-learning framework for predicting pattern dynamics on planar domains, spherical surfaces, and general manifolds. The method decom- poses the forward evolution operator into a long-range component, represented by a compact Fourier multiplier constructed via the Sum-of-Exponentials (SOE) approximation, and a short-range component adapted to the un- derlying geometry and its intrinsic symmetries. For general manifolds represented by irregularly sampled point clouds, the long-range component is implemented by Gaussian gridding onto an auxiliary regular grid, where the Fourier multiplier is efficiently applied in k-space using FFT and the result is interpolated back to the original sample points. We evaluate LSR-Net on several benchmark systems, including the Allen–Cahn, Cahn–Hilliard, Schnakenberg, and Turing systems, over planar domains, spherical surfaces, and a blob-shaped manifold. Numer- ical results demonstrate that LSR-Net consistently achieves higher accuracy and improved stability compared with baseline operator-learning models. In particular, for Allen–Cahn dynamics on the sphere, the RMSE is reduced by approximately three orders of magnitude compared with the Spherical Fourier Neural Operator (SFNO). Rota- tion and reflection equivariance tests further confirm that the learned operator is consistent with these geometric transformations. These results indicate that LSR-Net provides an effective and robust approach for learning pattern dynamics on complex geometries.

ABSTRACT. In order to predict the long-term effects of irradiation on the material properties of tungsten, a continuum approach to simulating the interactions of dislocation loops, which arise from radiation damage, is proposed. Continuum models of the displacement, strain and stress fields produced by dislocation loops exhibit unphysical singularities near the defect core, but are thought to accurately capture atomistic displacements in the far-field. A linear elastic model of nanoscale dislocation loops in tungsten is developed, and the model is verified using atomistic simulations to ensure that the model is informed by lower-length scale phenomena such that the physics of the problem is correctly captured. We discuss the model and its advantages, and show that predictions produced by atomistic simulations do indeed agree well with the far-field behaviour of the continuum model when dislocation loops are far from material boundaries. In particular, we robustly demonstrate that the decay rate of atomistic results and continuum results coincide with one another, and show that the results converge as the size of the atomistic simulations approach the far-field limit

ABSTRACT. We propose an explicit fully discrete scheme for approximating the invariant measure, and hence the ergodic limit, of the stochastic Burgers--Huxley equation driven by additive noise. The method combines a parameterized strongly tamed accelerated exponential Euler integrator in time with a spectral Galerkin discretization in space. To handle the simultaneous presence of the Burgers-type convection term and the polynomial reaction term, we identify a sharp parameter boundary for the coupled monotonicity property of the drift, which provides the analytical basis for the long-time analysis. Under this condition, we establish uniform-in-time moment bounds for the numerical solution in $L^\infty$-norm and prove that the fully discrete scheme admits a unique invariant measure. We then analyze the associated Kolmogorov equation and derive a time-uniform weak convergence result with explicit spatial and temporal rates of $\lambda_N^{-\beta+\epsilon}+\tau^{\beta-\epsilon}$, where $\beta\in(0,1]$ characterizes the regularity of the noise, including the case of space--time white noise. These weak rates are essentially sharp in the sense that they are twice the corresponding strong convergence orders. Moreover, this weak error estimate, together with the uniqueness of the numerical invariant measure, leads to an explicit error bound for approximating the invariant measure of the exact equation by the numerical invariant measure, and hence for approximating the associated ergodic limit.These results provide a rigorous and computationally efficient approach to long-time simulation, ergodic approximation, and invariant-measure approximation for stochastic reaction--diffusion--convection equations beyond the standard reaction--diffusion setting.

ABSTRACT. The Magnus expansion provides a powerful tool for solving linear, time-dependent differential equations of the form $X'(t) = A(t) X(t)$. It expresses the solution as an exponential of an infinite series whose terms involve iterated integrals of nested commutators of the operator $A(t)$ evaluated at different times. In practice, truncating this series yields approximate solutions, but quantifying the resulting error is critical. In this work, we introduce a general procedure to derive explicit, rigorous error bounds for these approximations in the unitary case. Crucially, our method expresses these error estimates directly in terms of the norms of these nested commutators, thus exhibiting commutator scaling. Additionally, we show that these bounds can also be formulated in terms of the norm of the operator $A(t)$. Finally, we illustrate the practical utility and accuracy of these new estimates across several applied settings.

ABSTRACT. Connections on principal bundles are central objects in differential geometry and appear naturally in gauge field theories in physics. Yang–Mills connections, the critical points of the Yang–Mills action functional, are of particular interest. Computing these on non-trivial bundles requires a discretization that reflects the topological structure of the bundle.

We present a finite element method for computing Yang–Mills connections on principal bundles over compact Riemannian manifolds. Given a triangulation of the base manifold and local sections of the bundle over each simplex, the connection form is represented piecewise. The transfer functions between overlapping sections prescribe tangential and normal jump conditions across interfaces. These jumps encode the topology of the bundle in a way that is amenable to finite element discretization.

In the case of an abelian structure group, the jump conditions become linear and can be enforced weakly through Lagrange multipliers, leading to a saddle-point problem over broken Sobolev spaces of differential forms. We characterize the associated jump and dual spaces, prove well-posedness via inf-sup conditions, and derive error estimates with optimal convergence rate using the trimmed polynomial spaces from finite element exterior calculus.

The poster will also present a numerical experiment on the Hopf bundle over the two-sphere, where the Yang–Mills connections have known analytical representations. The implementation uses the Firedrake library, and the results confirm the predicted convergence rate.

ABSTRACT. In this poster we will present some results on a phase field topology optimization problem for 3D printing applications. The model involved is a coupled system with Cosserat continuum mechanics and thermal spectral dissipation. We establish well-posedness and differentiability of the state systems, adjoint systems and prove the existence of minimizers. Furthermore, we derive the first-order optimality conditions and provide a second-order sufficient condition to characterize local minima. This is a joint work with Kei Fong Andrew, Lam and Ehsan-Ul-Haq (HKBU).

ABSTRACT. The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE constraints have been widely used to deal with this problem. However, such pproaches typically require repeated iterations and solving the forward problem at each iteration, which leads to a heavy computational cost. To address this issue, we first reformulate the inverse conductivity problem as a minimization problem involving a regularized residual functional. We then transform this minimization problem into a variational problem and establish the equivalence between them. This reformulation enables the employment of the finite element method to reconstruct the shape of the object from finitely many measurements. Notably, the proposed approach allows us to identify the object directly without requiring any iterative procedure. A prior error estimates are rigorously established to demonstrate the theoretical soundness of the finite element method. Based on these estimates, we provide a criterion for selecting the regularization parameter. Additionally, several numerical examples are presented to verify the feasibility of the proposed approach in shape reconstruction.

ABSTRACT. In this poster, we consider the solution of the steady flow in a three-dimensional lid-driven cavity using numerical methods. The three-dimensional velocity-vorticity formulation, used by Davies & Carpenter (2001), is considered. The cubical lid-driven cavity problem is solved. The problem is discretized using the Chebyshev discretization in the y and z directions, and fourth-order finite differences are used for the discretization in the x direction. Newton linearization is used to linearize the problem and a direct solver is devoted to solve the problem. The problem has been coded in both the MATLAB and FORTRAN environments. The lid-driven cavity problem is used typically to test new methods and codes. The lid-driven cavity can be introduced as a fluid contained in a cube domain with stationary rigid walls and a moving wall.

ABSTRACT. Fractional differential equations have found application across a wide range of engineering disciplines, where they can capture non-local physical behaviour that integer-order models do not adequately represent. However, simulating such models is computationally demanding, due primarily to the fractional derivative’s inherent non-locality. This work presents the implementation of fractional differential equations in the modelling of engineering systems. Using an established multi-step product-integration trapezoidal method, numerical solutions for large nonlinear dynamical systems which involve multiple timescales are considered. Known computational challenges encountered in this implementation are highlighted. For example, as the numerical method uses the entire solution history, the demands on computer memory are considerably greater than for integer-order solvers. Aspects that require particular care to achieve accurate and meaningful results are discussed, including the treatment of initial conditions and the use of known analytical expressions for fractional derivatives of specific functions within the model. The impact of including fractional derivatives is illustrated through sample results. This work is offered as a contribution to the discussion of applying numerical methods for solving fractional differential equations to engineering systems.

ABSTRACT. This paper presents and analyzes a novel fast two-grid alternating direction implicit (ADI) numerical scheme for efficiently solving two-dimensional nonlinear time-fractional reaction-diffusion equations involving the Caputo derivative of order $\mu \in (0,1)$. To handle the weak initial singularity in the solution, a fitted mesh is employed, while the time-fractional derivative is discretized using a high-order convergent sum-of-exponentials technique, significantly reducing both storage requirements and computational cost. Furthermore, an ADI strategy is developed to solve the resulting two-dimensional system by splitting it into two successive one-dimensional subproblems. To further enhance computational efficiency, a spatial two-grid technique is incorporated, where a nonlinear problem is solved on a coarse grid followed by a linear problem on a fine grid. A rigorous stability and convergence analysis is established through a discrete energy method, local truncation error estimates, and the fractional Gr\"onwall inequality, proving unconditional stability and optimal convergence rates. Numerical experiments validate the theoretical results and demonstrate the efficiency and effectiveness of the proposed fast two-grid ADI scheme.

ABSTRACT. We study the free energy and dynamics of a closed elastic filament (a one-dimensional curve in two dimensions) whose local internal state is specified by curvature, stretch, and a scalar density field representing, for example, the concentration of an absorbed species. The density variable has a tendency to phase-separate whereas the local spontaneous curvature is concentration-dependent. There is also a coupling between concentration and the stretching of the filament, although our main interest is in the nearly inextensible regime. We formulate and simulate the dynamics, comprising a coupled Willmore flow and Cahn–Hilliard gradient flow on the full differential geometry of a closed filament, addressing issues that previous work typically sidestepped by restricting to the Monge gauge. We use a numerical strategy for global free energy minimization, presenting the equilibrium shapes and density profiles across a wide range of model parameters. The phase diagram is dominated by a relatively small number of simple shapes that exhibit, as expected, strong coupling between local curvature and concentration. We also find regimes where curvature and/or stretching energies suppress phase separation altogether. For selected parameter values we present fully dynamical results, tracking the time evolution of the various contributions to the free energy. The dynamics often arrive at metastable minima rather than the equilibrium state -- for example, at states with more than the minimum number of interfaces between coexisting phases. The metastability of these states is absent for phase separation on a rigid circular domain and thus a direct result of the coupling between geometry and density.

ABSTRACT. Machine Learning Potentials have revolutionized computational chemistry by bypassing the O(N^3) computational bottleneck of first-principles Density Functional Theory. Within this domain, Kernel Ridge Regression (KRR) and Gaussian Process Regression (GPR) offer exceptional data efficiency and rigorous mathematical extrapolation. However, to maintain physical symmetries, atomic coordinates must be encoded into high-dimensional local chemical environment descriptors, such as SOAP. For complex multi-element systems, the descriptor dimensionality can rapidly escalate to thousands. Consequently, evaluating the exact kernel matrix for predicting atomic forces imposes an overwhelming computational and memory complexity of O(NMd). This research proposes a novel, theoretically optimal acceleration scheme leveraging Quasi-Monte Carlo slicing and the Non-equispaced Fast Fourier Transform to achieve near-linear scaling without sacrificing the essential physical smoothness of the predicted force field.

ABSTRACT. We propose a numerical method to construct transport mappings $\boldsymbol{m} = \nabla u$ between two sets $\mathcal{X},\mathcal{Y}$, with prescribed positive mass densities $f\in L^1(\mathcal{X})$ and $g\in L^1(\mathcal{Y})$. The class of mappings is defined through a decomposition of $\mathcal{X}$ in two subdomains, each of which is mapped onto $\mathcal{Y}$, resulting in a two-to-one transport structure. The resulting formulation can be interpreted as a mixed-type Monge-Ampère problem, where the Jacobian determinant changes sign across the subdomains. In contrast with classical results from optimal transport, where convexity of $u$ ensures some properties such as uniqueness and injectivity of $\boldsymbol{m}$, the proposed approach allows for a structured non-injective behavior while maintaining the gradient structure of $\boldsymbol{m}$. We introduce a least-squares scheme to approximate solutions of this system. Computational experiments indicate that the method effectively matches the prescribed densities with global convergence as a function of the spatial discretization.

ABSTRACT. Rare transition events in metastable systems under noisy fluctuations are crucial for many non-equilibrium processes, where the primary contributions to reactive flux are near the transition pathways between two metastable states. Efficient computation of these paths remains numerical challenges particularly in high dimensions. In this work, we examine the temperature-dependent maximum flux path, the minimum energy path, and the minimum action path at zero temperature and propose the StringNET method for training these paths by building the variational formulations. Unlike traditional nudged elastic band and string method driven by the tangent-projected gradient force, StringNET seeks the intrinsic variational formulation in path space and parametrizes the paths by neural network functions. We show that the loss function for the maximum flux path serves as a softmax approximation to the numerically challenging minimax problem of the minimum energy path, and develop the pre-training strategy that includes the maximum flux path loss in the early training stage, significantly accelerating the computation of minimum energy and action paths. We demonstrate the superior performance of this method through various analytical and chemical examples, as well as the two- and four-dimensional Ginzburg-Landau functional energy.

ABSTRACT. We present different strategies to decouple the equations of poroelasticity. This includes semi-explicit as well as iterative approaches.

ABSTRACT. This poster presentation analyses on passive vibration mitigation by Nonlinear Energy Sinks (NESs) coupled to a harmonically forced linear primary oscillator. Building on the classical complexification–averaging framework, three generalisations of the canonical cubic NES are analysed in a unified manner. For each configuration the slow-flow ODEs, the slow invariant manifold (SIM) equation, the fold/saddle–node conditions and the criterion for the strongly-modulated-response (SMR) regime are derived in closed form. The three cases studied are: (i) an essentially non-linear NES with arbitrary odd-power stiffness showwn,; (ii) an NES with linear stiffness but odd-power damping of NES, in which the SIM is proved to lose its S-shape and SMR is structurally absent; (iii) a hybrid NES combining cubic stiffness and odd-power damping.

ABSTRACT. Moving contact lines arise when one fluid displaces another on a solid substrate. While their modelling and simulation have been widely studied, most previous work has focused on rigid substrates. In this talk, we investigate the case of highly deformable elastic sheets. We first consider the capillary folding of thin elastic sheets with pinned contact lines. Using a three-dimensional model that combines interfacial energy with nonlinear shell elasticity, we study the relaxation dynamics and equilibrium shapes, and obtain numerical results that capture a range of experimentally observed folding behaviours. In the second part, we consider the motion of a thin liquid film on a highly bendable elastic sheet. Under lubrication approximation, we derive coupled evolution equations for the fluid interface and the sheet deformation. A matched asymptotic analysis in the limit of vanishing slip length yields an extended Cox-Voinov relation linking contact line speed to apparent contact angle. The results show that substrate deformability significantly influences wetting dynamics.

ABSTRACT. We consider a prototypical SPDE with finite-dimensional multiplicative noise which, subject to a nonnegative initial datum, admits a unique nonnegative solution. Inspired by well-established techniques from the deterministic setting, we introduce a finite element discretization of this SPDE that is convergent and which, for nonnegative initial data preserves nonnegativity of the numerical solution throughout the evolution. This part is joint work with C. Le Bris and E. Süli.

In addition, we introduce a fully discrete numerical method that combines mass-lumped finite elements with a Lie–Trotter splitting strategy. This discretization preserves nonnegativity at the discrete level and is shown to be convergent under suitable regularity conditions. This part is joint work with C. Le Bris and O. Hearder. Finally, we comment on the challenges arising in extending these techniques to the Dean–Kawasaki equation with the goal of preserving positivity.

ABSTRACT. The numerical solution of implicit and stiff differential equations by implicit numerical integrators has been largely investigated and there exist many excellent efficient codes available in the scientific community, as Radau5 (based on a Runge-Kutta collocation method at Radau points) and Dassl, based on backward differentiation formulas, among many others.

When solving time fractional problems, the derivative operator is replaced by a non-local one and it is reformulated as a Volterra integral equation (of convolution type), to which these codes cannot be directly applied.

The main idea of our numerical approach is to approximate the fractional kernel by a sum of exponential functions, and then to transform the fractional integral (of convolution type) into a set of ordinary differential equations. Solving the resulting (stiff) differential equation by a numerical time integrator gives a fast and memoryless numerical method.

It is explained how the code RADAU5 can be used for solving fractional differential equations. Numerical experiments illustrate the accuracy and the efficiency of the proposed method.

ABSTRACT. Semi-discretization of parabolic problems often leads to large and stiff systems of ODEs. Standard explicit methods then require very small time steps for stability, while implicit methods allow larger steps but involve solving linear systems. Explicit stabilized methods provide an efficient compromise by enlarging the stability region while remaining fully explicit. However, when local spatial mesh refinement is introduced, their efficiency decreases, since the stiffness is driven by only the smallest mesh element. A natural approach is to split the system into fast stiff and slower mildly stiff components. In this context, Abdulle, Grote, and Rosilho de Souza (2022) proposed the first-order multirate explicit stabilized method (mRKC). We extend their approach to second order and introduce the new multirate ROCK2 method (mROCK2), which achieves high precision and allows a step-size strategy with error control.

ABSTRACT. Splitting methods are a natural choice for the numerical time integration of partial differential equations, and arbitrary high order splitting schemes exist for Schrödinger equations with periodic boundary conditions. However, in the presence of non-periodic boundary conditions, we show that they suffer in general from an order reduction, even for smooth initial conditions. In this talk, we introduce a family of modified splitting methods for one-dimensional linear Schrödinger equations with Dirichlet boundary conditions, which achieve an arbitrary high order, and do not suffer from such order reduction phenomena. This is illustrated with a fourth order splitting scheme considering initial conditions with various regularity properties.

ABSTRACT. We present deep density methods for nonlinear Bayesian filtering in discretely observed, continuous-time SDEs that remain accurate and feasible in high dimensions. Starting from the classical prediction–update recursion, the (unnormalized) filtering density evolves between observations by a Fokker–Planck PDE and is updated at observation times by Bayes’ rule. In high dimensions, densities become extremely small, making direct density learning unstable. We therefore derive a log-density formulation, through a reverse Cole–Hopf transformation, of the filtering PDE, yielding a robust objective and improved numerical behaviour in high-dimensional regimes. This setting is naturally posivity-preserving and our numerical scheme inherits this property.

Computationally, we connect the PDE recursion to a nonlinear Feynman–Kac representation, yielding a Forward Backward SDE form of the prediction step. We approximate the resulting forward–backward systems with the deep BSDE method, producing a filter that can be trained offline and deployed online for rapid sequential inference. Under a parabolic Hörmander condition, we establish a hybrid error estimate with a computable residual.

We benchmark the method against a deep splitting filter and classical baselines (extended and ensemble Kalman filters, particle filters) across linear and nonlinear examples, including partially observed Lorenz–96 up to dimension 100. In this regime, particle-based methods degrade while the log-density deep methods remain stable. Empirically, we observe two to five orders of magnitude faster inference time than particle filters in the high-dimensional setting.

ABSTRACT. We derive first-order bounds on the bias in Wasserstein distances of the invariant measure of stochastic gradient kinetic Langevin dynamics with minimal assumptions on the stochastic gradient noise. These bounds sharpen existing non-asymptotic guarantees for stochastic-gradient MCMC methods and provide a quantitative resolution of a previously open problem on invariant measure accuracy. The main technical ingredients are new Gaussian convolution inequalities controlling the Wasserstein-p distance between a Gaussian convolved with a mean-zero perturbation and the Gaussian itself. We anticipate that these inequalities will be of independent interest beyond the present application.

ABSTRACT. We consider Langevin stochastic differential equations (SDEs) with reflecting boundary conditions. This is a setting where the interaction of stochastic dynamics with the boundary plays an important role. One of the major applications of such SDEs is constrained sampling. We develop numerical schemes which efficiently incorporate reflection at the boundary within the approximating Markov chain. We analyze the schemes to obtain the optimal order of convergence and present numerical experiments supporting the theoretical results.

ABSTRACT. In this presentation, we consider the numerical approximation of the stochastic complex Ginzburg-Landau equation with additive noise on the one-dimensional torus. The complex nature of the equation means that many of the standard approaches developed for stochastic partial differential equations cannot be directly applied. Our method is based on a spectral discretization in space and a Lie-Trotter splitting method in time. We obtain moment bounds for the numerical method before proving our main result: strong convergence on a set of arbitrarily large probability. From this, we also obtain the convergence in probability. In this talk, I will highlight the specific challenges posed by the stochastic complex Ginzburg-Landau equations, particularly in contrast with stochastic evolution equations featuring non-globally Lipschitz terms, such as the stochastic Allen-Cahn equation.

ABSTRACT. Numerical approaches to treat higher order nonlinear Schrödinger equations are presented. We study ground states of the equations, their stability and potential blow-up of the solutions.

ABSTRACT. Despite its applications in models such as Alfvén waves and water waves, numerical methods for the derivative nonlinear Schrödinger (dNLS) equation are relatively scarce due to regularity loss issues caused by derivative nonlinearities. In this talk, we present a first-order, unfiltered exponential integrator for the one-dimensional dNLS equation with low-regularity initial data. For any $s>1/2$, the method converges with first order in time in $H^s(\mathbb{T})$ for initial data $u_0\in H^{s+1}(\mathbb{T})$, losing only one derivative. Moreover, we construct a symmetrized version that improves the global error behavior and exhibits better conservation properties.

ABSTRACT. We propose and analyze a twice-filtered resonance-based exponential-type integrator (FREI) for the ``good'' Boussinesq (GB) equation. By combining two carefully designed filters with the discrete Bourgain framework, we establish rigorous error bounds for solutions with very low regularity. In particular, the method converges for initial data $(\phi_0,\psi_0)\in H^s\times H^{s-2}$ with $0<s\le1$. For a suitable choice of the spatial truncation parameter $K$ depending on $s$, the numerical scheme achieves an $L^2$-error bound of order $\mathcal{O}(\tau^{r(s)})$, where $r(s)=s/2$ for $0<s\le1/4$ and $r(s)=s^2/3+s/2$ for $1/4<s\le1$. This represents a clear improvement over the classical $\mathcal{O}(\tau^{s/2})$ convergence of single-filtered methods for rough data. Numerical experiments confirm the theoretical results and demonstrate the efficiency and robustness of the proposed approach.

ABSTRACT. We study splitting schemes for the time integration of the 3D energy-(sub)critical semilinear wave equation on the full space and the torus under the finite-energy condition. In the case of a cubic nonlinearity, we show that a filtered Strang splitting converges with almost second order in L^2 and almost first order in H^1. If the nonlinearity has a quartic form instead, we show analogous convergence results, where the order is reduced by 1/2 in both cases. For the energy-critical quintic nonlinearity, we show first-order convergence in L^2 for the filtered Lie and Strang splittings. To our knowledge these are the best convergence results available for the 3D semilinear wave equation under the finite-energy condition and they include the first error analysis performed for a scaling-critical problem. Notably, our analysis for the critical defocusing problem on the full space is even global in time. Our approach relies on continuous- and discrete-time Strichartz estimates. Dispersive estimates in discrete time were already used in the context of semilinear Schrödinger equations by Ignat, Ostermann, Schratz, and others. We also make use of the integration and summation by parts formulas to exploit cancellations in the error terms. Moreover, in the torus case, error bounds for a full discretization using the Fourier pseudo-spectral method in space are given. Finally, we discuss a numerical example indicating the sharpness of our theoretical results. This is partially joint work with Roland Schnaubelt (Karlsruhe).

ABSTRACT. (joint with A. Aimi, T. Chaumont-Frelet, G. Di Credico, C. Guardasoni, I. Labarca Figueroa, J. Nick) We survey recent work on space and space-time adaptive mesh refinement procedures for Galerkin and convolution quadrature discretizations of wave equations, formulated as an equivalent boundary integral equation. Reliable a posteriori error estimates of residual type lead to adaptive boundary element methods, based on the four steps: Solve - Estimate - Mark - Refine. We discuss the a posteriori analysis and the resulting space and space-time adaptive Galerkin elements, as well as extensions to convolution quadrature discretizations. Numerical experiments confirm the theoretical results. They study the convergence and computational efficiency of the approaches and the reliability and efficiency of the error estimates.

ABSTRACT. In this work in progress we are interested in solving numerically the mixed boundary value problem for the wave equation on non-cylindrical domains, based on the boundary integral equation techniques. We introduce related time-domain boundary integral operators, and their representation formulas; one of the key differences with the classical, cylindrical case, comes from the new definition of the conormal derivative. Next, we concentrate on the Dirichlet problem in the one-dimensional case, discuss the mapping and stability properties of the single-layer potential and single-layer boundary integral operator. Our analysis is largely based on energy techniques. For the single-layer boundary integral equation, we present a convolution quadrature inspired discretization. The originality lies in the fact that the resulting equations are no longer of convolution form. Finally, we present numerical experiments illustrating our findings.

ABSTRACT. In this talk we revisit the coupling of time-domain boundary integral equations with domain methods in first order form. We will show how to obtain stable discretisations without the need for any stabilisation parameters required in earlier formulations. We will show that both acoustic and Maxwell equations fit the setting as well as nonlinear and dynamic boundary conditions. The effectiveness of the formulation and its discretisation will be illustrated by numerical experiments.

ABSTRACT. In this talk, we consider a scattering problem in full space from an obstacle which is penetrable but governed by a nonlinear material law. Mathematically, this is modeled by coupling the free-space linear wave equation in the (unbounded) exterior domain with a semilinear wave equation inside of the (bounded) scatterer. The discretization is then carried out by coupling a finite element method with timestepping for the nonlinear part with a convolution quadrature based boundary integral discretization in the unbounded domain. We present a rigorous a-priori analysis of the fully discrete scheme and talk about the challenges inherent in this problem.

ABSTRACT. In this talk, we study optimization on networks modeled as metric graphs. Motivated by applications where the objective function is expensive to evaluate or only available as a black box, we develop Bayesian optimization algorithms that sequentially update a Gaussian process surrogate model of the objective to guide the acquisition of query points. To ensure that the surrogates are tailored to the network's geometry, we adopt Whittle-Matérn Gaussian process prior models defined via stochastic partial differential equations on metric graphs. In addition to establishing regret bounds for optimizing sufficiently smooth objective functions, we analyze the practical case in which the smoothness of the objective is unknown and the Whittle-Matérn prior is represented using finite elements. Numerical results demonstrate the effectiveness of our algorithms for optimizing benchmark objective functions on a synthetic metric graph and for Bayesian inversion via maximum a posteriori estimation on a telecommunication network.

ABSTRACT. Machine learning has thrived by training expressive models on large datasets, but is now shifting toward integrating prior knowledge—like physical laws or structure—into the learning process. Many such problems can be formulated as an implicitly constrained learning (ICL) task, where a prediction model is subject to a partially known constraint whose unknown component is inferred by aligning model predictions with data. ICL problems arise across domains: enforcing physics in scientific models, planning from a learned world model, or accounting for hidden variables for causal effect estimation.

These settings pose major challenges, especially when using expressive over-parameterized models like deep neural networks: (1) the resulting optimisation can become ill-defined; (2) current methods (e.g., differential programming, penalization approaches, neural operators) are often unstable, inaccurate, or inefficient; (3) learning theory for ICL is largely under-developed, limiting understanding of generalization and robustness.

In this talk, I present a methodological framework for solving ICL, capable of handling complex implicit constraints—ranging from optimality conditions to partial differential equations and dynamical systems. The approach adopts a functional viewpoint that reframes ICL as a well-defined yet abstract problem, using deep networks only as approximation tools. We apply this framework to a class of bilevel optimization problems in Hilbert spaces covering several applications such as instrumental variable regression and model-based reinforcement learning. We expect this functional viewpoint on ICL to extend beyond Hilbert spaces to cover measure spaces, with possible applications to generative modeling, in particular for efficiently "guiding" diffusion models.

ABSTRACT. Variational regularization techniques are dominant in the field of inverse problems. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. This issue can be approached by machine learning where we estimate these parameters from data. This is known as ”Bilevel Learning” and has been successfully applied to many tasks, some as small-dimensional as learning a regularization parameter, others as high-dimensional as learning parameters of neural networks. While mathematically appealing, this strategy leads to a nested optimization problem which is practically challenging since function values and gradients cannot be computed to a high-enough precision. In this talk we discuss new computational approaches for this problem which do not assume exact knowledge of these. It turns out that a clever choice on the accuracy leads to much faster yet stable and robust solutions.

ABSTRACT. This work proposes a computer-assisted framework for systematically designing Lyapunov functions to analyze convergence rates of continuous-time dynamical systems associated with optimization algorithms. While many optimization methods such as gradient descent, accelerated gradient methods, and Newton-type flows can be interpreted as ordinary differential equations (ODEs), the construction of suitable Lyapunov functions for proving convergence rates has largely relied on heuristic and problem-specific insights.

Building on recent constructive approaches to Lyapunov design, we reformulate the remaining heuristic choices such as integration-by-parts manipulations and structural parameter selections as a finite symbolic search problem. Using computer algebra, we exhaustively explore candidate Lyapunov functions within a structured template and reduce the validity conditions to semidefinite constraints. This enables systematic verification and, in certain cases, optimization of convergence rates.

Our framework recovers known results for classical systems, including gradient flows and Nesterov-type dynamics under convexity and strong convexity assumptions, thereby validating the method. Moreover, it identifies improved convergence guarantees or alternative Lyapunov constructions in several cases, demonstrating that prior analyses may not have been exhaustive.

The proposed approach advances the systematization and semi-automation of Lyapunov analysis in optimization, reducing reliance on manual ingenuity and enabling reproducible, structured exploration of continuous-time models. It also opens the possibility of discovering new dynamical systems with desirable convergence properties through computationally guided design.

ABSTRACT. We develop dynamical low-rank (DLR) filtering approaches for data assimilation problems described by stochastic differential equations (SDEs). The proposed framework is fully online and particularly well suited for high-dimensional problems with intrinsic low-rank structure, a common feature in filtering applications. In particular, we construct a DLR filter that minimizes the local mean-squared error, as well as we provide a possible strategy to approximate the complementary directions. This approach allows to obtain a subspace which evolves also according to the observation operator. These approaches naturally reduce to a Kalman–Bucy-type filter in the case of linear drift. In more general settings, they lead to ensemble-type methods and are therefore applicable to systems with nonlinear drift and potentially non-Gaussian distributions. These methodologies can be extended to smoothing procedures in a suitable way. In addition, we introduce a preliminary particle-type DLR filter and a transport map DLR filter, both of which show promising performance in nonlinear regimes. Numerical experiments demonstrate the effectiveness of the proposed methods in representative applications.

[KMNZ2026] Yoshihito Kazashi, Youssef Marzouk, Fabio Nobile, and Fabio Zoccolan, “Dynamical Low-Rank Filters for Data Assimilation”, in preparation 2026

[MZ2026] Youssef Marzouk and Fabio Zoccolan, “Dynamical Low-Rank Smoothing”, in preparation 2026

ABSTRACT. Radiation therapy simulations are fundamentally limited by the curse of dimensionality, as particle densities must be resolved over high-dimensional phase spaces spanning position, direction, and energy, in combination with heterogeneous patient-specific materials. This results in rapidly growing computational complexity and memory demands as resolution is increased, making accurate simulations particularly challenging. In particular, achieving clinically relevant accuracy requires fine phase-space discretizations and careful treatment of strongly anisotropic scattering, often leading to prohibitive runtime and resource requirements. We address these challenges by employing dynamical low-rank approximation (DLRA) to reduce the effective dimensionality of the underlying transport problem while retaining essential physical features. In this talk, I discuss computational bottlenecks in current high-dimensional simulation frameworks for radiation therapy simulations and identify how these limitations can be leveraged to improve numerical accuracy rather than merely reduce cost. This is joint work with Pia Stammer, Steffen Schotthöfer, and Cory Hauck.

ABSTRACT. Kinetic equations are used to model a wide range of phenomena important for real-world applications. Their applications span astrophysics, nuclear physics, engineering, and social sciences. Due to their high-dimensional phase space, modelling and quantifying uncertainties, relevant for applications, poses a significant challenge even for modern computing infrastructure. In recent years, dynamical low-rank approximation (DLRA) has gained popularity for making fine grid simulations of high-dimensional problems feasible by evolving the solution of a time-dependent PDE as a low-rank factorization. This reduces the computational and memory requirements significantly.

In this work, we propose a low-rank multilevel Monte Carlo estimator for kinetic equations based on a probabilistic rank-adaptive DLRA time integrator. The level hierarchy of the low-rank multilevel estimator is constructed through spatial refinement of the domain and by ensuring that the low-rank error remains below the spatial discretization error. We demonstrate the efficacy of the estimator through several numerical experiments from radiation transport, radiation therapy, and shallow water flow.

ABSTRACT. Low Rank (DLR) methods are a promising way to reduce the computational cost and memory footprint of the high-dimensional thermal radiative transfer (TRT) equations. The TRT equations are a system of nonlinear PDEs that model the energy exchange between the material temperature and the radiation energy density. Due to their high dimensionality, solving the TRT equations is often a bottleneck in multi-physics simulations. DLR methods represent the solution in terms of time-evolving SVD-like factors of angle and space. Here we present our recently developed method that uses a discrete set of time-evolving angles. This DLR formulation enables us to use the highly optimized SN transport sweep as our main computational kernel, and thus results in a practical way of leveraging low-rank methods in scalable production TRT codes that are routinely run on large supercomputers. We demonstrate the methods on several challenging, highly heterogeneous problems in two spatial dimensions (4D), where these DLR schemes can give a significant reduction in angular artifacts (i.e., ray effects) with the same cost as gold-standard SN methods.

ABSTRACT. The main challenge in the analysis of numerical methods for the Zakharov system (ZS) originates from the presence of derivatives in the nonlinearity. In this talk, we present a novel reformulation of the ZS, which allows us to construct second-order time symmetric methods and higher-order numerical methods for the ZS even with generalized nonlinear terms. By considering exponential wave integrators (EWIs) for this reformulation, a new time symmetric EWI is formulated and its properties are studied.

ABSTRACT. We consider the numerical approximation of the nonlinear Klein–Gordon equation in the nonrelativistic limit regime, where the solution exhibits highly oscillatory behavior in time with wavelength of order O(\varepsilon^2). In this regime, classical numerical methods typically require small time steps to resolve the oscillations.

In this talk, we present a multiscale time integrator method for the NKGE, which is based on (i) a multiscale decomposition by frequency in each time interval with simplified transmission conditions, and (ii) an exponential wave integrator for temporal discretization and a Fourier pseudospectral method for spatial discretization.

We show that the proposed scheme achieves both optimal accuracy in space and uniformly first-order convergence rate in time. In addition, a multiscale interpolation in time is presented for obtaining a uniformly accurate approximation of the solution at any time t\ge0 by adopting a linear interpolation of the micro variables within the multiscale decomposition in each time interval. Numerical experiments illustrate the efficiency and robustness of the method compared to classical approaches.

This work provides a systematic framework for designing uniformly accurate schemes for highly oscillatory partial differential equations.

ABSTRACT. In this talk, we will introduce a multiscale method for the numerical solution of a highly oscillatory Stokes problem. This method is based on a technique called localized orthogonal decomposition (LOD), which enables approximation properties independent of the regularity of the coefficients. Applying the LOD to a reformulated Stokes problem allows us to construct exponentially decaying basis functions for the velocity approximation while using a piecewise constant pressure approximation. This reformulation is necessary to avoid conflicts between the global nature of the divergence-free constraint and our desire to identify rapidly decaying basis functions. The exponential decay motivates the localization of the basis computation, which is essential for the practical realization of the method. We perform a rigorous a priori error analysis and prove optimal convergence rates for velocity and pressure approximations provided that the support of the basis functions increases logarithmically with the desired accuracy. Numerical experiments confirm our theoretical findings.

ABSTRACT. Splitting methods are among the most powerful tools in the numerical treatment of highly oscillatory evolution problems, as they efficiently exploit the underlying structure of the vector fields involved.

More than a decade ago, work co-authored by the speaker, a symmetric Zassenhaus splitting (sZS) method was introduced for the semiclassical Schrödinger equation, where the solution exhibits severe oscillations induced by the small semiclassical parameter. The remarkable performance of the method was closely related to the structure of the repeatedly nested commutators appearing in the Zassenhaus expansion: these operators possess very small spectral radii, allowing their exponentials to be computed efficiently by Krylov subspace techniques with only a few Lanczos iterations.

However, the original analysis was formulated only for finite-dimensional operators and did not address regularity assumptions on the exact solution.

In this talk, we present a rigorous error analysis of the symmetric Zassenhaus splitting in the infinite-dimensional setting. We derive precise regularity assumptions ensuring the validity of the expansion and establish error bounds for general unbounded operators A and B.

In particular, the theory applies to highly oscillatory problems like semiclassical Schrödinger problems, providing a rigorous analytical foundation for the excellent numerical behaviour observed in practice.

ABSTRACT. Conjugate phase retrieval considers the recovery of a function, up to a unimodular constant and conjugation, from its phaseless measurements. In this talk, I will introduce the conjugate phase retrieval for complex-valued signals residing on graphs, and explore its applications to shift-invariant spaces. We first provide the sufficient condition for the conjugate phase retrieval of graph signals, construct two explicit graphs, and show that graph signals residing on them are conjugate phase retrievable. Next, we explore its applications for the signals living in the shift-invariant spaces, and study the conjugate phase retrievable of the shift-invariant signals from the structured phaseless samples.

ABSTRACT. Wavelets are sparse multiscale representation systems that have been used in various applications such as signal and image processing, numerical PDEs, and data science. In this talk, I will focus on their application to numerically solving two important problems in science and engineering: the Helmholtz equation, which models time-harmonic wave propagation, and elliptic interface problems. I will describe some challenges we often face in solving these two PDEs and present our wavelet Galerkin method as an effective solution. The benefits of using wavelets to address these challenges will be discussed, along with details on wavelet constructions, and key theoretical and computational results related to our method.

ABSTRACT. In this talk, we discuss the structures of the variational characterization of the spherical t-design, its gradient, and its Hessian in terms of fast spherical harmonic transforms. Moreover, we propose solving the minimization problem of the spherical t-design using the trust-region method to provide spherical t-designs with large values of t. Based on the obtained spherical t-designs, we develop (semi-discrete) spherical tight framelets and their fast spherical framelet transforms for practical spherical signal/image processing. Thanks to the large spherical t-designs and the localization property of our spherical framelets, we are able to provide signal/image denoising using local thresholding techniques based on a fine-tuned spherical cap restriction. Many numerical experiments are conducted to demonstrate the efficiency and effectiveness of our spherical framelets and spherical designs, including Wendland function approximation, ETOPO data processing, and spherical image denoising.

ABSTRACT. Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary. In a widely used algorithm, extended dynamic mode decomposition (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. We analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary’s nonlinear functions are homogeneous. In this presentation, we show that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deepDMD algorithm . We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

ABSTRACT. Transfer operators such as the Koopman and Perron-Frobenius operators provide a powerful linear framework for analyzing nonlinear dynamical systems, with their spectral properties revealing long-term behavior and coherent and metastable structures. Classical data-driven approximation methods like EDMD for the Koopman operator approximation rely on a predefined dictionary of basis functions whose choice is highly problem-dependent and often requires domain knowledge. We propose a randomized neural network method, called RaNNDy, for learning transfer operators and their spectral decompositions, in which hidden-layer weights are fixed at random and only the output layer is trained. This results in a significant computational advantage, closed-form expressions for the output layer of the neural network that directly represents the eigenfunctions, and enables uncertainty quantification via ensemble learning. The effectiveness of these methods is demonstrated on several examples, including SDEs, protein folding dynamics, and the quantum harmonic oscillator.

ABSTRACT. Solving time-dependent Partial Differential Equations (PDEs) accurately and efficiently remains a key challenge in computational science. Physics-Informed Neural Networks (PINNs) offer a mesh-free framework by embedding physical laws, expressed as PDEs, directly into the loss function, eliminating the need for costly simulation data. However, traditional PINNs are often slow to train and can be inaccurate, mainly due to their reliance on gradient-descent-based iterative optimization and their non-causal handling of time as an additional spatial dimension. In this talk, we introduce a significantly faster and more accurate alternative to conventional PINN training, termed Frozen PINNs. Our approach is based on the principle of space-time separation and combines random feature methods with classical ordinary differential equation solvers. We demonstrate, through results on several challenging PDEs, that Frozen PINNs can significantly outperform stochastic gradient descent. Their performance is even comparable to state-of-the-art mesh-based methods, such as isogeometric analysis, for low-dimensional problems, while also extending effectively to high-dimensional PDEs and avoiding the need for specialized hardware like GPUs.

ABSTRACT. The Koopman operator and its associated generator enable the derivation of global linear representations of general nonlinear dynamical systems. However, this typically requires embedding the dynamics into an infinite-dimensional space. Developing data-driven finite-dimensional approximations of the Koopman operator and generator has been a major focus of research over the past two decades. Approaches include the extended dynamic mode decomposition (EDMD), the generator EDMD and their extensions, such as kernel-based formulations and deep learning methods for dictionary learning. Conversely, when shifting from linear to polynomial representations of at least degree two, the existence of finite-dimensional embeddings is assured for a broad class of nonlinear dynamical systems. We will introduce methods for data-driven modelling of these systems that utilise the embedding-space formulation. Applications encompass system identification, surrogate modelling, and prediction. We will also draw connections to the Koopman formalism and sparse identification of nonlinear dynamics (SINDy).

ABSTRACT. Over the past decade, quantum computers have matured from experimental units with only a couple of qubits to hundreds of qubits with first implementations of error correction. Last year, breakthroughs in quantum computing networking have been announced as well. As such, we are rapidly approaching the point at which quantum networking will be part of the quantum computing landscape and classical security measures will no longer be appropriate.

In this talk, I will give an introduction the challenges faced when combining quantum communications channels with isogeny-based post-quantum cryptography. In particular, I will look at the specific challenge of onion routing in quantum networks. Onion routing is a method that not only keeps a message hidden, but also obfuscates who is talking to whom. While basic cryptographic ideas can be generalized to quantum networks, quantum onion routing has to contend with specific challenges arising from quantum computing restrictions.

ABSTRACT. It is well known that arbitrary-angle rotations can be implemented fault-tolerantly to arbitrary precision by compilation to a Clifford+T gate set. This approach is scalable, but is generally understood to have a high overhead of tens of T gates per rotation. In this talk, we will describe our work to reduce the cost of Clifford+T synthesis, and its application to significantly reduce the cost of fault-tolerant Trotterization for chemistry problems. Our main tool to investigate this task is taking probability and quasi-probability mixtures of Clifford+T channels. In particular, we show that the T cost to implement a rotation channel can be reduced significantly for small-angle rotations, and returning to existing angle-independent results in the worst case. This dispels the commonly-stated claim that Clifford+T synthesis has a high overhead independent of the rotation angle, and is particularly impactful for Trotter circuits, which are dominated by small-angle rotations.

Using our results, we show that the T-gate cost of Trotter circuits compiled to Clifford+T gates becomes constant in the limit of small Trotter step sizes, and can be reduced by orders of magnitude even for large step sizes. Our results are also important for early fault-tolerant quantum computers, as they allow rotations to be implemented with shorter Clifford+T sequences, and should allow more straightforward incorporation of error mitigation.

ABSTRACT. Many scientific-computing problems lead to dynamics that are not naturally unitary, including dissipative PDEs, stochastic differential equations, imaginary-time projection methods, and constrained differential-algebraic equations. This talk presents a universal dilation viewpoint for embedding such nonunitary dynamics into unitary or projected unitary dynamics on an enlarged Hilbert space.

I will present a moment-matching dilation framework for linear differential equations and its extensions to linear Itô SDEs, auxiliary-field quantum Monte Carlo, and constrained PDE dynamics. Applications include PDEs, stochastic trajectories, near-term AFQMC circuit implementations, and the unsteady Stokes equations.

The talk aims to highlight dilation as a flexible and broadly applicable framework for bringing nonunitary models from numerical analysis and scientific computing into quantum simulation.

ABSTRACT. We consider general probabilistic systems in which gates act linearly on some infinite-dimensional vector space and factor (densely) through a (compact, simple, connected, Lie) group. Thus, every state-effect pair gives a [0,1]-valued (probability) function on this group. We give a method to reconstruct the adjoint representation of the group, using only evaluations of gate sequences on the system. Our method works by constructing a finite-difference stencil matrix, where the rows are indexed by well-spread probe words and the columns are indexed by words approximating tangent directions near the identity. When the derivatives of the probability function at the probe points span the Lie algebra, conjugation queries then reveal how each gate rotates the tangent space, recovering the adjoint action through this embedding. The learned representation enables us to take any (long) gate sequence and replace it with a compiled short sequence (using Solovay-Kitaev like procedure) which will faithfully approximate output probability for any state and effect.

ABSTRACT. Data assimilation (DA) enables high-fidelity predictions of dynamical systems by integrating numerical models of the system with experimental measurements. In particular, classical adjoint-variational DA methods estimate the solution which most accurately reproduces a time series of sparse measurements, by optimising a constrained objective function for the initial condition of the solution. We propose to optimize in a lower-dimensional latent representation of the dynamical system which is learned by implicit rank minimizing autoencoders. This learned latent space is capable of identifying the physically meaningful perturbation directions which matter most for accurate estimation of the true initial condition. When applied to two-dimensional, monochromatically-forced turbulence (Kolmogorov flow), latent-space DA estimates the full turbulent state with a relative error improvement of two orders of magnitude over classical adjoint-variational DA. The small scales of the estimated turbulent field are predicted more faithfully with latent-space DA, greatly reducing erroneous small-scale velocities typically introduced by state-space DA. Furthermore, latent-space DA is robust to the corruption of the measurements with Gaussian noise, which may be introduced by imperfect sensors. These findings demonstrate that the observability of dynamical systems from available data can be greatly improved when the measurements are assimilated in the right space, or coordinates.

ABSTRACT. Deep learning-based methods have shown remarkable effectiveness in solving PDEs, largely due to their ability to enable fast simulations once trained. However, despite the availability of high-performance computing infrastructure, many critical applications remain constrained by the substantial computational costs associated with generating large-scale, high-quality datasets and training models. In this work, inspired by studies on the structure of Green's functions for elliptic PDEs, we introduce Neural-HSS, a parameter-efficient architecture built upon the Hierarchical Semi-Separable (HSS) matrix structure that is provably data-efficient for a broad class of PDEs. We theoretically analyze the proposed architecture, proving that it satisfies exactness properties even in very low-data regimes. We also investigate its connections with other architectural primitives, such as the Fourier neural operator layer and convolutional layers. We experimentally validate the data efficiency of Neural-HSS on the three-dimensional Poisson equation over a grid of two million points, demonstrating its superior ability to learn from data generated by elliptic PDEs in the low-data regime while outperforming baseline methods. Finally, we demonstrate its capability to learn from data arising from a broad class of PDEs in diverse domains, including electromagnetism, fluid dynamics, and biology.

ABSTRACT. In many scientific settings, acquiring clean data can be expensive, hazardous, or impossible. Recent diffusion-based methods can reconstruct clean data from corrupted measurements, but they require clean examples during training—a requirement that is often unrealistic in practice.

In this talk, we introduce Ambient Physics, a framework for learning the clean data distribution directly from corrupted observations, without requiring a single clean example. The key idea is to randomly mask a subset of already-observed measurements and supervise on them, so the model cannot distinguish “truly unobserved” from “artificially unobserved”, and must produce plausible predictions everywhere.

Despite never seeing clean training examples, Ambient Physics achieves state-of-the-art reconstruction performance. Compared with prior diffusion-based approaches, it reduces average overall error by 62.51% while using 125× fewer function evaluations. These results point toward a more practical paradigm for scientific machine learning: learning physics directly from corrupted data that real experiments produce.

ABSTRACT. In this talk I will present our recent progress on establishing rigorous first-of-its-kind upper bounds on the generalization error for the method of approximating solutions to the (d+1)-dimensional incompressible Navier-Stokes equations by training depth-2 neural networks trained via the unsupervised Physics-Informed Neural Network (PINN) framework. This is achieved by bounding the Rademacher complexity of the PINN risk. For appropriately weight bounded net classes our derived generalization bounds do not explicitly depend on the network width and our framework characterizes the generalization gap in terms of the fluid's kinematic viscosity and loss regularization parameters. In particular, the resulting sample complexity bounds are dimension-independent. Our generalization bounds suggest using novel activation functions for solving fluid dynamics. We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor-Green vortex benchmark.

Reference : https://arxiv.org/abs/2603.23072

ABSTRACT. High-energy particle beams are routinely used to heat plasmas in e.g. fusion experiments. It has been observed that the beam profile strongly influences heating efficiency. This naturally raises the question: which beam profile is optimal? Since the plasma dynamics is governed by a hyperbolic partial differential equation (the Vlasov equation) and the parameter space can be large, this naturally leads to a high-dimensional optimization problem. It turns out that the corresponding parameter landscape is extremely rugged with numerous local minima. Consequently, many classic optimization approaches do not perform well (e.g. gradient-based methods get easily stuck in local minima).

We compare a number of different methods (basin hopping, Bayesian-based surrogate models, variants of genetic algorithms, among others). In particular, we show that differential evolution performs well and is relatively robust. However, it still requires a large number of PDE solves. To reduce this computational cost, we use a rank-adaptive dynamical low-rank strategy to speed up the solution of the Vlasov equation. We demonstrate that the combination of differential evolution and low-rank methods significantly increases efficiency, while still allowing us to get very close to the optimal solutions.

ABSTRACT. We introduce a diagonal quasi-Newton optimiser called momentum diagonal BFGS (mdBFGS). This optimizer combines BFGS‑style curvature updates with exponential moving averages to form a per‑parameter preconditioner for large‑scale stochastic and non-convex problems. The preconditioner is updated via a damped secant condition evaluated on exponentially averaged curvature pairs, which reduces stochastic noise. A damping mechanism further stabilizes the preconditioner under mini‑batch gradients. The method is broadly applicable, from linear regression to large‑scale deep neural network training, and offers a simple, scalable alternative to both first‑order momentum methods and quasi‑Newton schemes.

ABSTRACT. Most optimal control problems (OCP) cannot be solved analytically, necessitating numerical approaches to obtain solutions for the state-adjoint variables and controls. Most methods fall into two categories: direct or discretise-then-optimise methods and indirect or optimise-then-discretise methods. In direct methods, one discretises the state dynamics and objective function and then solves the resulting finite-dimensional optimisation problem. In indirect methods, one applies Pontryagin’s maximum principle (PMP) to the continuous OCP to obtain necessary optimality conditions, which fully defines the control as a function of the state-adjoint variables, after which the state-adjoint dynamics are solved using a numerical integrator. We investigate the relation between these two numerical approaches, in the case of OCPs of second-order ODEs without control constraints. For this, we make use of a novel Lagrangian formulation of such OCPs. In this formulation, it is always possible to construct an indirect approach associated to a direct one, provided specific control discretisation and consistent state equations and cost function discretisation are imposed. Conversely, given an indirect method, an equivalent direct approach exists only if the state-adjoint equations are discretised using a symplectic method.

ABSTRACT. The turnpike property in optimal control refers to the behavior of optimal solutions to remain near a steady state for most of the time interval. It has recently been discovered that, when an optimal control problem admits symmetries, the turnpike is not static but instead corresponds to an orbit induced by the action of the symmetry group. This kind of turnpike is called a trim turnpike, and the limiting trajectory is given by a trim primitive. In this case, symmetry reduction of the state-adjoint system makes it possible to identify the limiting trajectory and then describe the convergence behavior of optimal solutions. This phenomenon is especially relevant for mechanical systems, where symmetries naturally arise and are related to conservation laws.

In this work, we analyze the influence of discretization on the preservation of the turnpike property. Our approach is based on two steps: the analysis of reduction in discrete state-adjoint systems, and the relation between the turnpikes arising from the reductions in the discrete and continuous settings. In addition, we study the relations between the rates of convergence of continuous and discrete solutions toward the corresponding turnpikes. We illustrate our findings with numerical examples, including examples from mechanics.

ABSTRACT. We present a parallel preconditioning framework for the iterative solution of huge-scale linear systems arising from time-dependent, locally interacting mean field games and Schroedinger bridges. Following a variational approach, we employ a finite difference discretization and solve the resulting finite-dimensional optimization problem using the Chambolle--Pock primal--dual algorithm. We embed within our solver a general class of parallel-in-time preconditioners based on diagonalization techniques in the time direction, implemented via discrete Fourier transforms. These enable efficient, parallel, scalable iterative solvers with robustness across a wide range of viscosities. For structured grids, we further develop fast recursive solvers that exploit tensor-product structure, while retaining flexibility for more general geometries.

ABSTRACT. Recently, continuous optimization methods based on the RSAV (relaxed scalar auxiliary variable) approach have been proposed, which ensure modified dissipation of gradient flows. Existing studies, however, rely solely on gradient information, and often exhibit degraded convergence when applied to ill-conditioned problems. In this work, we incorporate Hessian information into the linear operator arising in the RSAV scheme to accelerate convergence. Directly using the Hessian, however, presents two key challenges: (i) increased computational cost due to both the construction of the Hessian and the solution of the associated linear systems at each iteration; and (ii) potential loss of the modified dissipation property of the RSAV scheme when the Hessian is indefinite. To overcome these challenges, we propose using an approximate Hessian as the linear operator. To mitigate the first issue, we apply the Nyström method, a randomized low-rank approximation technique, to approximate the Hessian efficiently. To address the second, we enforce positive semidefiniteness of the approximate Hessian via eigenvalue decomposition. Numerical experiments on the training of PINNs (physics-informed neural networks), a class of typically ill-conditioned optimization problems, demonstrate that the proposed method yields accelerated convergence.

ABSTRACT. In our daily life, we can easily observe chemical reactions when digesting food, consuming energy and so on. Numerous energies are involved in chemical reactions, including free energy of reactants, products, and activation energy. Among them, the activation energy is the minimum amount of energy that should be needed to reactants to fire a chemical reaction. In other words, chemical reaction kinetics can be strongly affected by uncertainties in the activation energies, even when reaction free energies are well characterized. We study how perturbations in activation energies propagate through reaction rate equations under the assumption of thermodynamically consistent forward and reverse reactions.

Uncertainty quantification together with Arrhenius analysis based on peak reaction rates demonstrates the influence of activation energy uncertainty on reaction dynamics and its temperature dependence.

ABSTRACT. In real Hilbert spaces, given a single-valued Lipschitz continuous and monotone operator, we study generalized split feasibility problem (GSFP) over solution set of monotone variational inclusion problem. An inertia iterative method is proposed to solve this problem, by showing that the sequence generated by the iteration converges strongly to solution of GSFP. As against previous methods, our step size is chosen to be simple and not depending on norm of associated bounded linear map as well as Lipschitz constant of the single-valued operator. The obtained result was applied to study split linear inverse problem, precisely, the LASSO problem. Lastly, with the aid of numerical examples, we exhibited efficiency of our algorithm and its dominance over other existing schemes.

ABSTRACT. The Fock exchange operator constitutes a cornerstone of Hartree-Fock (HF) theory, playing a pivotal role in accurately describing quantum mechanical exchange effects essential for predicting electronic structures in molecules and condensed matter systems. However, its nonlocal nature introduces prohibitive computational costs, particularly in large-scale applications. This report introduces a generalized framework for constructing approximate Fock exchange operators in HF theory, addressing the computational bottlenecks caused by the nonlocal nature of the exact operator. By employing low-rank decomposition and incorporating adjustable variables, the proposed method ensures high accuracy for occupied orbitals while maintaining Hermiticity and structural consistency with the exact operator.

ABSTRACT. In this talk we describe a technique to avoid the order reduction that appears when integrating reaction-diffusion initial boundary value problems with explicit exponential Rosenbrock methods. These methods are an efficient tool to integrate nonlinear stiff differential systems when information on the Jacobian of the vector field which defines the differential system is available. Moreover, as an advantage with respect to explicit exponential Runge-Kutta methods, as the Jacobian is known at each step, we can achieve methods with a desired accuracy with less stages.

However, in the numerical integration of initial boundary value problems, they suffer from order reduction, as other exponential methods. When considering vanishing boundary conditions, this order reduction has been previously avoided by considering stiff order conditions on the coefficients of the method. Here, we describe a technique to avoid this order reduction with any exponential Rosenbrock method without having to impose stiff order conditions. Moreover, both the technique and the theoretical results are valid for general time-dependent boundary conditions. The suggested technique consists of discretizing firstly in time, by substituting the exponentials of operators applied over functions by initial boundary value problems for which suitable boundaries must be proposed. Besides, when trying to get local order up to four, we give a simplification of the suggested boundaries in order to calculate them as easily as possible without losing order. We also show some numerical experiments, corroborating the theoretical results, as well as the efficiency with respect to applying the standard method of lines.

ABSTRACT. Barren plateaus constitute a fundamental obstacle in the training of variational quantum algorithms (VQAs), particularly when applied to large-scale discretizations of nonlinear partial differential equations. In this talk, we introduce a domain decomposition framework designed to mitigate barren plateaus by localizing the cost functional. The spatial domain is partitioned into overlapping subdomains, each associated with a localized parameterized quantum circuit and measurement operator. This structure enables the optimization to be performed on smaller subsystems while maintaining consistency across the global domain. Numerical experiments for the time-independent Gross–Pitaevskii equation show that the domain-decomposed formulation, in which subdomain iterations are interleaved with optimization steps, achieves improved solution accuracy and more stable optimization behavior compared to a global VQA formulation.

ABSTRACT. Fluorescence molecular tomography (FMT) is a non-invasive optical imaging technique widely used in biomedical research, but it remains challenging due to severe ill-posedness and poor depth resolution. Conventional iterative methods often struggle to recover accurate z-axis information, while supervised learning methods rely on large paired training datasets that are difficult to obtain in practice. In this talk, we introduce a warm-basis iterative projection method (WB-IPM) that combines learning-based prediction with iterative refinement. The method uses the normalized neural-network output as a learned warm-basis direction and refines the remaining correction component through an iterative projection scheme. This design helps preserve useful learned depth information while improving reconstruction stability and accuracy. Theoretical analysis shows that the method mainly depends on the directional alignment between the ground truth and the neural-network output, motivating a cosine-similarity-based training loss. Simulated and real-data FMT experiments demonstrate improved reconstruction accuracy and robustness compared with standalone learning-based and iterative methods.

ABSTRACT. Recently, the escalating demand for memory and computational resources by large models has presented formidable challenges for their deployment in resource-constrained environments. One prevalent way to serve large models is to develop effective model compression approaches that crop the size of large models while maintaining acceptable performance levels. Currently, the landscape of model compression methodologies predominantly revolves around four fundamental algorithms: pruning, quantization, knowledge distillation, and low-rank approximation whose basic frameworks have been established years ago. However, the compression efficacy of these methods is often capped or challenging to scale based on theoretical analysis. In this talk, we introduce a novel and general-purpose approach, referred to as hyper-compression, that redefines model compression as a problem of parameter representation. Specifically, we extend the concept of hypernets into what we term a ‘hyperfunction'. Then, the hyperfunction is designed based on ergodic theory (ET). This advanced formulation leads to a performant algorithm that offers several distinct advantages, succinctly summarized as PNAS: 1) Preferable compression ratio; 2) No post-hoc retraining; 3) Affordable inference time; and 4) Short compression time. Lastly, in light of the observed stagnation in hardware Moore's Law, we conjecture “Moore's Law of Model Compression", i.e., the efficiency of model compression could double annually in the near future to meet the needs of large model era. We believe that model compression based on hyperfunction can play an important role in “Moore's Law of Model Compression".