Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN)

TL;DR

This paper introduces CRVPINN, an efficient collocation-based method that enhances the robustness and convergence speed of variational physics-informed neural networks for solving PDEs.

Contribution

It proposes a novel LU factorization approach for sparse Gram matrices in VPINNs, improving robustness and training efficiency.

Findings

CRVPINN achieves faster convergence in PDE solutions.

The method effectively handles Laplace, advection-diffusion, and Stokes problems.

Robustness is improved by translating continuum norms to the discrete level.

Abstract

Physics-Informed Neural Networks (PINNs) have been successfully applied to solve Partial Differential Equations (PDEs). Their loss function is founded on a strong residual minimization scheme. Variational Physics-Informed Neural Networks (VPINNs) are their natural extension to weak variational settings. In this context, the recent work of Robust Variational Physics-Informed Neural Networks (RVPINNs) highlights the importance of conveniently translating the norms of the underlying continuum-level spaces to the discrete level. Otherwise, VPINNs might become unrobust, implying that residual minimization might be highly uncorrelated with a desired minimization of the error in the energy norm. However, applying this robustness to VPINNs typically entails dealing with the inverse of a Gram matrix, usually producing slow convergence speeds during training. In this work, we accelerate the implementation of RVPINN, establishing a LU factorization of sparse Gram matrix in a kind of point-collocation scheme with the same spirit as original PINNs. We call out method the Collocation-based Robust Variational Physics Informed Neural Networks (CRVPINN). We test our efficient CRVPINN algorithm on Laplace, advection-diffusion, and Stokes problems in two spatial dimensions.