Collocation-based robust variational physics-informed neural networks (CRVPINNs)
Marcin Łoś, Tomasz Służalec, Paweł Maczuga, Askold Vilkha, Carlos Uriarte, Maciej Paszyński
Abstract
Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations (PDEs) through strong residual minimization formulations. Their extension to weak scenarios via Variational PINNs (VPINNs) has been shown to lack robustness when the discrete and continuous-level norms are mismatched. Robust Variational PINNs (RVPINNs) address this problem by appropriately incorporating the Gram matrix but suffer from high computational costs due to the weak residual integration and the Gram matrix inversion. In this work, we accelerate RVPINN computations by using a point-collocation approach similar to PINNs, and by employing an LU factorization of the sparse Gram matrix. This leads to the proposed Collocation-Based Robust Variational PINN (CRVPINN). We validate CRVPINN on Laplace, advection–diffusion, Stokes, non-linear stationary Navier–Stokes, and linear elasticity problems in two spatial dimensions, demonstrating improved efficiency without compromising robustness.