Robust Variational Physics-Informed Neural Networks

TL;DR

This paper presents a robust variant of Variational Physics-Informed Neural Networks (VPINNs) that minimizes a residual-based loss to improve error estimation and robustness in solving PDEs, demonstrated through advection-diffusion problems.

Contribution

It introduces a new residual-based loss functional for VPINNs that enhances robustness and provides reliable error estimates, supported by theoretical analysis and numerical experiments.

Findings

Loss based on discrete dual norm improves error estimation.

Algorithm demonstrates robustness across advection-diffusion problems.

Numerical results confirm sharpness of theoretical estimates.

Abstract

We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov-Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN's loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection-diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp.