Robust Variational Physics-Informed Neural Networks

TL;DR

This work introduces Robust Variational Physics-Informed Neural Networks (RVPINNs), which minimize the discrete dual norm of the weak residual by using the Riesz representative of the residual in the discrete test space. By selecting an appropriate test-space inner product and solving a local Galerkin problem to obtain a single test function, the method yields a loss that is provably equivalent to the true energy-norm error (up to oscillations under Fortin-type assumptions) and is robust to the choice of test-basis. The authors prove a posteriori and a priori error estimates, and demonstrate the approach on diffusion-advection problems, including 1D/2D smooth cases, delta-source forcing, and advection-dominated regimes, highlighting the method's stability and reliability where standard VPINNs struggle. They also discuss practical considerations such as Gram-matrix inversion, orthonormal test bases, and potential offline strategies for parametric problems, pointing to broad applicability and future extensions to broader PDE classes.

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.

Figures (11)

• Figure 1: RVPINNs approximation of the smooth diffusion problem; strong BCs imposition, $50$ spectral test functions.
• Figure 2: RVPINNs approximation of the smooth diffusion problem; strong BCs imposition, $100$ FE test functions.
• Figure 3: RVPINNs approximation of the 1D smooth diffusion problem; strong BCs imposition, $5$ FE test functions.
• Figure 4: RVPINNs approximation of the 2D smooth problem; strong BCs imposition, $20 \times 20$ spectral test functions.
• Figure 5: RVPINNs approximation of the 2D smooth problem; strong BCs imposition, $100 \times 100$ elements mesh with piece-wise linear FEM test functions.

Theorems & Definitions (28)

• Remark 1: The weak residual
• Remark 2: Existence of solutions for the Petrov-Galerkin-type discretization
• Remark 3: Classical approach
• Remark 4: Constraint assumption
• Remark 5: Riesz representative of the weak residual
• Remark 6: Gram matrix inversion
• Proposition 1
• proof
• Theorem 2: Error class bounds in terms of the residual representative
• proof