Python library supporting Discrete Variational Formulations and training solutions with Collocation-based Robust Variational Physics Informed Neural Networks (DVF-CRVPINN)

TL;DR

This paper introduces a Python library for solving PDEs using discrete weak formulations and neural networks, emphasizing robustness and error control in training solutions for problems like Stokes and Laplace equations.

Contribution

The paper presents a novel Python environment that combines discrete weak formulations with neural network training, including mathematical proofs of robustness and well-posedness.

Findings

Successfully trained neural networks for 2D Stokes equations using discrete weak residuals.

Demonstrated robustness of the loss function related to true error.

Extended the approach to Laplace problem formulation.

Abstract

We explore the possibility of solving Partial Differential Equations (PDEs) using discrete weak formulations. We propose a programming environment for defining a discrete computational domain, introducing discrete functions defined over a set of points, constructing discrete inner products, and introducing discrete weak formulations employing Kronecker delta test functions. Building on this setup, we propose a discrete neural network representation, training the solution function defined over a discrete set of points and employing discrete finite difference derivatives in the automatic differentiation procedures. As a challenging computational model example, we focus on Stokes equations in two-dimensions, defined over a discrete set of points. We train the solution using the discrete weak residual and the Adamax algorithm with discrete automatic differentiation of the discrete gradients. Despite introducing the python environment, we also provide a rigorous mathematical formulation based on discrete weak formulations, proving the well-posedness and robustness of the loss function. The solution of the discrete weak formulations is based on neural network training employing a robust loss function that is related to the true error. In this way, we have a robust control of the numerical error during the training of the neural networks. Besides the Stokes formulation, we also explain the functionality of the proposed library using the Laplace problem formulation.