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PINNsFailureRegion Localization and Refinement through White-box AdversarialAttack

Shengzhu Shi, Yao Li, Zhichang Guo, Boying Wu, Yang Zhao

TL;DR

This work introduces WbAR, a white-box adversarial attack refinement strategy for physics-informed neural networks (PINNs), designed to locate and refine regions where PDE solutions fail. By performing gradient-guided adversarial sampling via PINN-PGD and iteratively retraining, WbAR focuses training on high-residual areas, capturing complex features such as multi-scale behavior and sharp or oscillatory solutions. The authors prove that adversarial search can outperform random walks and derive an explicit bound on the generalization error under WbAR, validating the method theoretically. Extensive numerical experiments across 2D and high-dimensional Poisson equations, Burgers’ equation, diffusion, Allen–Cahn, and multiscale problems demonstrate that WbAR robustly locates failure regions and achieves superior accuracy compared with baseline sampling strategies, highlighting its potential as a general tool for solving complex PDEs with PINNs.

Abstract

Physics-informed neural networks (PINNs) have shown great promise in solving partial differential equations (PDEs). However, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To precisely and adaptively locate the critical regions that fail in the solving process we propose a sampling strategy grounded in white-box adversarial attacks, referred to as WbAR. WbAR search for failure regions in the direction of the loss gradient, thus directly locating the most critical positions. WbAR generates adversarial samples in a random walk manner and iteratively refines PINNs to guide the model's focus towards dynamically updated critical regions during training. We implement WbAR to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, high-dimensional Poisson equations, and Burgers equation with sharp solutions. The results demonstrate that WbAR can effectively locate and reduce failure regions. Moreover, WbAR is suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution.

PINNsFailureRegion Localization and Refinement through White-box AdversarialAttack

TL;DR

This work introduces WbAR, a white-box adversarial attack refinement strategy for physics-informed neural networks (PINNs), designed to locate and refine regions where PDE solutions fail. By performing gradient-guided adversarial sampling via PINN-PGD and iteratively retraining, WbAR focuses training on high-residual areas, capturing complex features such as multi-scale behavior and sharp or oscillatory solutions. The authors prove that adversarial search can outperform random walks and derive an explicit bound on the generalization error under WbAR, validating the method theoretically. Extensive numerical experiments across 2D and high-dimensional Poisson equations, Burgers’ equation, diffusion, Allen–Cahn, and multiscale problems demonstrate that WbAR robustly locates failure regions and achieves superior accuracy compared with baseline sampling strategies, highlighting its potential as a general tool for solving complex PDEs with PINNs.

Abstract

Physics-informed neural networks (PINNs) have shown great promise in solving partial differential equations (PDEs). However, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To precisely and adaptively locate the critical regions that fail in the solving process we propose a sampling strategy grounded in white-box adversarial attacks, referred to as WbAR. WbAR search for failure regions in the direction of the loss gradient, thus directly locating the most critical positions. WbAR generates adversarial samples in a random walk manner and iteratively refines PINNs to guide the model's focus towards dynamically updated critical regions during training. We implement WbAR to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, high-dimensional Poisson equations, and Burgers equation with sharp solutions. The results demonstrate that WbAR can effectively locate and reduce failure regions. Moreover, WbAR is suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution.
Paper Structure (25 sections, 5 theorems, 60 equations, 12 figures, 4 tables, 2 algorithms)

This paper contains 25 sections, 5 theorems, 60 equations, 12 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works and contributions
  3. Organization
  4. Preliminaries
  5. Adversarial attacks
  6. PINNs
  7. Collocation points sampling strategies and iterative training
  8. White-box Adversarial Attack Refinement for PINNs
  9. Motivation
  10. PINN-PGD
  11. WbAR
  12. Sampling Strategies Comparison
  13. Convergence Analysis of WbAR
  14. Numerical Experiments
  15. Two-dimensional Poisson equation
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose Assumption ass:r(ii) is satisfied. Let $r_{k-1}(x;\theta^*)$ be the model residual obtained after $k-1$ training iterations, the empirical risk of samples $\{s^{(n)}_k\}_{n=1}^{N_k}$ will be greater or equal to the empirical risk of samples $\{\tilde{s}^{(n)}_k\}_{n=1}^{N_k}$, i.e.,

Figures (12)

  • Figure 1: The training framework of PINN
  • Figure 2: The residual and samples of (a) a sharp solution, (b) an oscillatory solution, (c) a multi-peak solution problem.
  • Figure 3: High-dimensional Poisson equation experiment results
  • Figure 4: Burgers' Equation
  • Figure 5: Multiscale problem experiment results
  • ...and 7 more figures

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Theorem 5
  • proof