A physics-informed deep learning approach for solving strongly degenerate parabolic problems

TL;DR

This paper introduces a physics-informed deep learning method using PINNs to effectively solve strongly degenerate parabolic PDEs, demonstrating promising accuracy in multi-dimensional gas filtration problems.

Contribution

It is among the first to apply PINNs to strongly degenerate parabolic problems, showing their effectiveness in complex, real-world applications.

Findings

PINNs accurately predict solutions in 2D and 3D domains.

The approach estimates approximation errors effectively.

Numerical results validate the method's applicability to degenerate PDEs.

Abstract

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning frameworks for solving initial-boundary value problems involving nonlinear PDEs. Recently, PINNs have shown promising results in several application fields. Motivated by applications to gas filtration problems, here we present and evaluate a PINN-based approach to predict solutions to strongly degenerate parabolic problems with asymptotic structure of Laplacian type. To the best of our knowledge, this is one of the first papers demonstrating the efficacy of the PINN framework for solving such kind of problems. In particular, we estimate an appropriate approximation error for some test problems whose analytical solutions are fortunately known. The numerical experiments discussed include two and three-dimensional spatial domains, emphasizing the effectiveness of this approach in predicting accurate solutions.