2024/01/04 by Łoś, Marcin, Służalec, Tomasz, Maczuga, Paweł +3 · 1 citation
#65M99 #68T07 #FOS: Computer and information sciences #FOS: Mathematics #G.1.10 #G.1.8 #I.2 #I.m #J.2 #Machine Learning (cs.LG) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2401.02300
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.