Studying approximation spaces for neural networks (NNs) is of great importance, not only theoretically, but also for practical applications, such as designing, generalizing, and improving the performance of the NNs. In this paper, NNs were built based on multi-channel convolution and the concept of a tensor product of matrices. The concept of generalized and strict NNs was introduced. Also, the basic properties of the parameters in artificial NNs were studied, and present the classes of NNs to identify the properties that facilitate the study of NNs. Moreover, the definition of approximation spaces for generalized and strict NNs is presented, and it is proven that the classes of approximation spaces are quasi-Banach spaces and achieve continuous embedding. Therefore, the main goal of studying NNs is to determine the range of functions from quasi-Banach spaces, where approximation spaces provide an overview of the goal by identifying its properties and relationships. Finally, a multi-convolutional neural network model was built and trained on a representative dataset using a reduced loss function, with the mean squared error (MSE) used as a performance evaluation metric. The results showed a significant decrease in error, reflecting the model's ability to learn and achieve high approximation accuracy, generating good agreement between actual and expected values. This highlights the effectiveness of neural networks as a powerful tool for addressing mathematical modeling and numerical approximation problems.