ABSTRACT. Full-waveform inversion (FWI) has emerged as a powerful technique to achieve high-resolution subsurface imaging, yet its applications for viscoelastic media are an active area of research due to the associated computational challenges. In this work, we use a finite element approach to implement the forward and inverse processes for viscoelastic FWI on isotropic media using Firedrake. We formulate viscoelastic FWI as a topology optimization-based inverse problem by defining the spatial distribution of material parameters as design variables. The attenuation behavior for viscoelastic media is governed by the Kelvin-Voigt material model whereas Clayton-Engquist boundary conditions are applied to suppress artificial boundary reflections and approximate an unbounded domain. Two benchmark synthetic models are used to investigate the validity of the proposed framework both in terms of performance and accuracy, whereas the initial guess is considered to be homogeneous throughout the domain. The cost function (J_cost) of the inverse problem is defined as the least-squares misfit (L2 norm) between the wavefield data from synthetic and initial models. To further address the inherent ill-posedness of the inverse problem and to ensure convergence, J_cost is augmented with Tikhonov regularization. Another feature of this framework is that the gradient of J_cost with respect to design variables is calculated efficiently using discrete adjoint-state sensitivities obtained through automatic differentiation (AD) in Firedrake. AD provides precise gradients even without manually deriving and coding sensitivity kernels. Furthermore, the resultant non-linear optimization problem is solved with IPOPT (Interior Point OPTimizer), which provides a stable and efficient convergence to the desired design variables. The results of the study indicate that the framework can effectively extract the spatial distribution of viscoelastic parameters from surface measurements for both synthetic models. The results demonstrate the potential of numerical finite element tools in association with advanced optimization solvers for complex geophysical imaging problems.

ABSTRACT. Elastic Full Waveform Inversion (FWI) remains a challenging problem due to its significant computational cost in three-dimensional settings and the complexity introduced when adding different absorbing boundary conditions. In this work, we present the improved spyro code, an originally acoustic-only FWI software with added elastic propagation built using Firedrake which highlights how the UFL’s (Unified Form Language) high-level abstractions can significantly help with development and experimentation. We present initial results investigating the impact of cells-per-wavelength parameter choice in mesh design, mass-lumped elements, and absorbing boundary conditions on both forward and inverse accuracy and cost, and discuss challenges associated with deployment on different hardware architectures.

The project is sponsored by TotalEnergies according to Agência Nacional do Petróleo RD&I Clause (Resolution No. 918/2023).

ABSTRACT. One of the grand challenge problems in fusion is modelling plasma in the edge region of a tokamak. In this edge region, hot plasma interfaces with the reactor wall and cold neutral gas resulting in a computationally expensive multi-scale problem that is considered an exascale challenge. This plasma use case requires discretisations that represent distributions over both physical space and over the product of physical space and velocity space.

In this we talk we describe NESO-Particles (NP), our particle framework, and VANTAGE-Reactions, our Reactions framework. NP is a particle framework which utilises SYCL as a performance portability abstraction for CPU and GPU architectures. VANTAGE-Reactions builds on NP to provide an abstract framework for describing reactions between particle species in plasma.

ABSTRACT. This talk presents the first implementation of moving particles in Firedrake. This contribution is an abstraction for ODE schemes describing particle movement, realised by composing Firedrake's existing interpolation machinery with Lagrangian algorithms for tracing particles through mesh cells. A key novelty of this abstraction is the use of geometric pullbacks to compute particle movement in reference space which substantially simplifies the required mesh entity collision detection algorithms. Building on this, we present a prototype particle trajectory solver and discuss how this initial work lays the groundwork for more advanced models coupling particle dynamics and continuum fields, such as kinetic-fluid simulations in fusion plasmas.

ABSTRACT. Physical units matter in scientific computing, yet most finite element frameworks lack built-in dimensional analysis. We present an approach to integrating physical units into the Unified Form Language (UFL) via a symbolic `Quantity` class that tracks dimensions within variational forms.

A graph-based visitor automates consistency checks and unit factorization across the expression tree. We also show that the resulting automated nondimensionalization acts as a physics-aware diagonal preconditioner.

Numerical examples include improved conditioning for Navier–Stokes system, detection of floating-point cancellation in Neo-Hooke hyperelasticity at small deformations, and handling of coupled multiphysics scaling in a Poisson–Nernst–Planck system. The implementation targets FEniCSx but the concepts apply directly to any UFL-based framework, including Firedrake.

The functionality is implemented in the [`dolfiny` framework](https://github.com/fenics-dolfiny/dolfiny).

ABSTRACT. Solutions of coupled multiphysics problems are crucial for modelling complex geophysical systems. These systems consist of multiple (potentially non-conforming) meshes, with the physics on each mesh governed by some PDE. The coupled interactions between each mesh can then be captured using cross-mesh interpolation operators.

Firedrake had previously been limited to the assembly of these operators only between CG/DG finite element spaces. Further, the parallel scaling performance of the assembly had been suboptimal.

This talk will present an overview of how these operators are constructed, how recent work has made feasible the assembly of these operators in massively parallel applications, and how Firedrake has been extended to assemble these operators between between non-conforming Hcurl and Hdiv finite element spaces.

ABSTRACT. Wavetanks or experimental basins are widely used in the maritime industry to model offshore and nearshore environments through scaled physical testing. These facilities typically include controllable wavemakers at one end and wave absorption mechanisms—such as sloping beaches—at the other, effectively minimising unwanted reflections in the offshore zone. To better reflect this physical setup, we present a numerical wavetank coupled with the surf zone at the beach. The two-dimensional model partitions the domain into deep- and shallow-water regions governed by nonlinear potential-flow and nonlinear shallow-water equations, respectively. Coupling conditions are derived from a unified variational principle, and the two subdomains are solved using a finite-element approach and a well-balanced finite-volume scheme, respectively. In a representative test case, over 97% of the total wave energy is dissipated by the absorbing beach. This extension, along with performance improvements using Firedrake “Constants”, enables the simulation of wave energy dissipation through shoaling, cresting and breaking, resulting in a cost-effective platform for modelling the full life cycle of water waves—from generation and propagation to interaction and dissipation. An outstanding question is whether and how a more automated variational derivation and implementation with further speed-ups can be facilitated within Firedrake. This work and presentation was funded by the EU (GA859983) and the Leeds Institute for Fluid dynamics, and was published in Lu et al. (2026), cf. Lu (2025) and Gidel (2018).

Y. Lu, F. Gidel, T. Bunnik, O. Bokhove, M. Kelmanson 2026: Variational and numerical water-wave and surf-zone hydrodynamics. J. Eng. Math. 157. https://doi.org/10.1007/s10665-026-10511-9

Y. Lu 2025: Numerical wavetanks for wave generation, interaction, and dissipation: variational and computational modelling. PhD dissertation, University of Leeds. https://etheses.whiterose.ac.uk/id/eprint/37486/

F. Gidel 2018: Variational water-wave models and pyramidal freak waves. PhD dissertation, University of Leeds. https://etheses.whiterose.ac.uk/id/eprint/21730/

ABSTRACT. The Geoscientific ADjoint Optimisation PlaTform (G-ADOPT — https://gadopt.org) is a next-generation Firedrake application, a computational framework for simulating geoscientific flows. Developed and maintained by researchers at the Research School of Earth Sciences, Australian National University, in collaboration with international partners, G-ADOPT delivers accurate, efficient, flexible, and extensible open-source software for both forward and inverse, data-driven geoscientific simulations (Davies et al. 2022; Ghelichkhan et al. 2024; Scott et al. 2026). By building on Firedrake's high-level abstractions, automated code generation, and pyadjoint integration, G-ADOPT inherits the scalability, transparency, and reproducibility needed for modern high-performance computing environments, while exposing domain-specific functionality to geoscientists.

Current applications span geodynamics (including mantle convection and its diverse surface expressions), glacial isostatic adjustment and sea-level change, and groundwater modelling. These problems typically demand large-scale HPC resources, with simulations routinely executed on more than 1000 cores.

This presentation will provide an overview of recent developments in G-ADOPT, its applications, and capabilities enabled by Firedrake. There will be a focus on ongoing development efforts and an outline of the opportunities for collaboration on core Firedrake functionality to support the next generation of geoscientific models.

ABSTRACT. Tidal stream energy is poised for significant growth, with contracts for 140 MW of capacity in the UK already secured. Strengthening investor confidence in the technology and its integration in the energy supply mix is predicated on accurate predictions of available resource and effective strategies for optimising energy extraction. Achieving these goals requires precise calibration of coastal ocean models and robust optimisation tools for array siting. In shallow-water equation modelling of tidal dynamics, bottom friction is the primary input parameter used to calibrate models against elevation and velocity measurements. The scarcity of available measurements and their spatial concentration in specific areas means that relatively satisfactory agreement between modelled and measured data over typical campaign durations (usually around one month) can be achieved with a practically unlimited range of uniform or spatially varying bottom friction fields. Each of these fields would respond differently to the introduction of turbines, which in turn can lead to sub-optimal array design. To reduce uncertainty in model predictions, adjoint-based optimisation tools for regularisation and data assimilation are actively being developed within the Firedrake-based Thetis coastal ocean model. Recent advancements have extended these methods from idealised scenarios to calibration of bottom friction fields in real-world sites using multi-decadal measurement datasets, allowing for calibration with unprecedented data volume and accuracy. Development has also begun on tools for studying how competing design and operational strategies affect the performance of arrays within the same coastal region – a growing concern in the marine energy sector. The talk will present recent advances in adjoint-based tool development in coastal ocean modelling, discuss challenges related to their implementation and outline plans for future work.

ABSTRACT. Over the last decade, the Met Office has been developing Momentum: a new atmospheric model, for numerical weather prediction and climate projection. The motivation was to be able to exploit the exascale generation of supercomputers by developing more scalable methods, and this primarily meant moving away from the latitude-longitude mesh to the cubed-sphere mesh, which is quasi-uniform over the sphere. The new model is underpinned by compatible finite element methods, which facilitate the non-orthogonal mesh whilst maintaining many desirable numerical properties.

Momentum is now nearing operational deployment. This talk will summarise the design of Momentum, and the role of finite element methods within it. I will also outline potential future model improvements, and the role Firedrake has played in developing these new improvements.

ABSTRACT. We investigate flows of incompressible fluids with Navier's slip boundary condition. Since the slip parameter is difficult to measure directly we investigate ways to determine this parameter from available flow data. For example, modern 4D-PC MRI velocity reconstruction data can be used in data assimilation procedure to estimate the Navier's slip parameter. This is achieved by using Firedrake adjoint framework and tested on artificially generated data. We examine how discretization impacts the overall robustness and efficiency of the proposed approach.

ABSTRACT. 4DVar data assimilation uses real-word observations to improve the accuracy of simulations, and is commonly used in operational weather forecasting. 4DVar involves solving an inverse problem for a timeseries which best matches the observations, constrained by a PDE model, using Gauss-Newton iterations. 4DVar requires a PDE model, observation operators, forward and adjoint derivatives, and efficient Gauss-Newton preconditioners. Recently proposed parallel-in-time preconditioners for “weak constraint” 4DVar add yet another layer of complexity.

We present fdvar, a library for parallel-in-time 4DVar data assimilation of any PDE in Firedrake (https://www.firedrakeproject.org/fdvar/). fdvar uses Firedrake’s automatic differentiation capabilities to automate the construction of the 4DVar system and it’s derivatives, and provides a variety of time-parallel preconditioners for solving the Gauss-Newton iterations using TAO, PETSc’s optimisation library.

fdvar will enable: application scientists to quickly implement and test 4DVar for their application; numerical analysts to easily trial novel models and solvers; and performance profiling with a fully space-time parallel implementation.

ABSTRACT. Modelling complex physical systems through partial differential equations (PDEs) is central to many disciplines in science and engineering. However, in most real applications, missing physics within the PDE model, expressed as unknown or incomplete relationships, such as constitutive or thermal laws, limits the description of the physics of interest. Existing surrogate modelling approaches aim to address this knowledge gap by learning the PDE solution directly from data, and in some cases, by also adding known physical constraints. However, these approaches are tailored and tied to specific system configurations (e.g., geometries, boundary conditions, or discretisations) and do not directly learn the missing physics, but only the PDE solution. We introduce FEML, an end-to-end differentiable framework for learning missing physics that combines the PDE modelling of the system of interest (known physics) with ML modelling of the operator representing the missing physics. By embedding a PDE solver into training, our approach allows one to train such operators directly from the PDE solution, which unlocks the learning of unknown relationships when the operator output cannot be directly measured (e.g., stress signals for learning constitutive laws). FEML dissociates the PDE modelling, which is tied to the system configuration considered, from the operator representing the missing physics, which is agnostic to system configurations and common across all physical systems that share the same physical properties (i.e., the hidden physics). Consequently, our framework naturally allows for zero-shot generalisation of complex physical systems that share the same hidden physics. It also enables downstream study of the learned model by domain specialists. Our framework uses structure-preserving operator networks (SPONs) to model the missing physics operator, which allows one to preserve key continuous properties at the discrete level, to learn over complex geometries and meshes, and to achieve zero-shot generalisation across different discretisations (i.e., different mesh resolutions and/or FE discretisations). We showcase our framework and its versatility by progressively recovering the elastoplastic constitutive law of an unknown material: first learning the nonlinear elastic response from force measurements, then freezing the learned elastic operator and discovering the plastic hardening law from full-field displacement data. The two pretrained operators are combined into a foundation constitutive model and applied zero-shot to a three-dimensional torsion problem. We further identify temperature-dependent conductivity in transient heat flow and show that symbolic regression applied to the learned neural operator recovers a closed-form power law matching the analytical structure of the ground truth.

ABSTRACT. Thermochemical energy storage (TCES), where thermal energy is stored in a reversible chemical reaction in a porous powder bed, is a promising technology for large-scale and long-term thermal energy storage. However, the scaling up of TCES reactors is hindered by the limited heat transfer from the heat source to the powder bed. Therefore, heat conducting structures, such as fins, are incorporated into the powder bed to enhance thermal contact. This talk will present recent advances in the numerical topology optimization of these heat conducting structures using Firedrake and machine learning. Due to the prohibitive time requirements of direct simulations, an artificial neural network surrogate model was used for the optimization. The model was trained on simulated data, generated with random fin structures. Then, the trained network is used to predict the progression of the reaction inside the reactor over time. In this presentation, we will present the most recent findings on the use of neural networks based on the SinGAN architecture for surrogate modeling. Further, we will show recent results for the topology optimization. The presentation will show the methodology used to couple the surrogate model with topology optimization algorithms, which are based on the brute force, level-set, and stochastic optimization methods. We show the usage of these methods to calculate optimal geometries for heat-conducting structures minimizing different objective functions, which encode the desired reactor performance characteristics.