PINN-DG: Residual neural network methods trained with Finite Elements

TL;DR

This paper introduces a novel class of Physics-Informed Neural Networks trained with finite element methods, improving efficiency and stability in solving PDEs by avoiding costly derivative computations and leveraging finite element advantages.

Contribution

It presents a new PINN framework integrated with discontinuous Galerkin finite element methods, offering theoretical convergence analysis and enhanced numerical performance.

Findings

Improved computational efficiency over standard PINNs.

Enhanced robustness and stability in PDE solutions.

Theoretical convergence guarantees.

Abstract

Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical techniques such as finite elements and finite differences. In this paper, we introduce a new class of Physics-Informed Neural Networks (PINNs) trained using discontinuous Galerkin finite element methods. Unlike standard collocation-based PINNs that rely on pointwise gradient evaluations and Monte Carlo quadrature, our approach computes the loss functional using finite element interpolation and integration. This avoids costly pointwise derivative computations, particularly advantageous for elliptic PDEs requiring second-order derivatives, and inherits key stability and accuracy benefits from the finite element framework. We present a convergence analysis based on variational arguments and support our theoretical findings with numerical experiments that demonstrate improved efficiency and robustness.