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

Marcin Łoś, Tomasz Służalec, Paweł Maczuga, Askold Vilkha, Carlos Uriarte, Maciej Paszyński

TL;DR

This work accelerates 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, and calls out method the Collocation-based Robust Variational Physics Informed Neural Networks (CRVPINN).

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.

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

TL;DR

This work accelerates 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, and calls out method the Collocation-based Robust Variational Physics Informed Neural Networks (CRVPINN).

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.
Paper Structure (16 sections, 6 theorems, 113 equations, 16 figures)

This paper contains 16 sections, 6 theorems, 113 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Abstract framework
  3. Numerical results for the Collocation method for Robust Variational Physics Informed Neural Networks
  4. Two-dimensional Laplace problem with sin-sin right-hand side
  5. Two-dimensional Laplace problem with exp-sin right-hand side
  6. Two-dimensional advection-diffusion problem
  7. Poisson problem with varying diffusion function
  8. Poisson problem with a jump
  9. Stokes problem with manufactured solution
  10. Discrete Stokes formulation
  11. Stability of discrete Stokes formulation
  12. Robust loss for Stokes problem
  13. Summary
  14. Computational costs
  15. Conclusions
  16. ...and 1 more sections

Key Result

Lemma 1

Given $u,v\in D_{0,h}$, it satisfies

Figures (16)

  • Figure 1: Sparsity pattern of the Gram matrix ${\bf G}$ build with $H^1_0$ norm.
  • Figure 2: Convergence of CRVPINN and the true error $H^1_0(\Omega_h)$ for the Laplace problem with sin-sin right-hand side.
  • Figure 3: Solution obtained from CRVPINN for the Laplace problem with sin-sin right-hand side.
  • Figure 4: Convergence of CRVPINN and the true error $H^1_0(\Omega_h)$ for the Laplace problem with sin-exp right-hand side.
  • Figure 5: Solution obtained from CRVPINN for the Laplace problem with sin-exp right-hand side.
  • ...and 11 more figures

Theorems & Definitions (12)

  • Lemma 1: Discrete integration by parts
  • proof
  • Lemma 2: Discrete product rule
  • proof
  • Lemma 3: Discrete norm equivalence
  • proof
  • Lemma 4: Boundedness of $b$
  • proof
  • Lemma 5: Coercivity of $b$
  • proof
  • ...and 2 more