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Incorporating Local Hölder Regularity into PINNs for Solving Elliptic PDEs

Qirui Zhou, Jiebao Sun, Yi Ran, Boying Wu

TL;DR

This work tackles solving elliptic PDEs with physics-informed neural networks by embedding local Hölder regularity as a regularization term in the loss. The authors introduce a local Hölder seminorm-based loss, approximated via a variable-distance sampling scheme, and derive a generalization bound linking the augmented loss to the approximation error and quadrature inaccuracies. Under standard stability assumptions, the resulting error estimate shows the solution discrepancy $\|u_\theta-u^*\|_{2,\alpha;\Omega}$ is controlled by the augmented loss and sampling error. Numerical experiments on second-order ODEs, Poisson, variable-coefficient elliptic, and Helmholtz problems demonstrate substantial improvements in accuracy and error distribution compared to standard PINNs, highlighting practical benefits for PDE surrogates and robustness to perturbations.

Abstract

In this paper, local Hölder regularization is incorporated into a physics-informed neural networks (PINNs) framework for solving elliptic partial differential equations (PDEs). Motivated by the interior regularity properties of linear elliptic PDEs, a modified loss function is constructed by introducing local Hölder regularization term. To approximate this term effectively, a variable-distance discrete sampling strategy is developed. Error estimates are established to assess the generalization performance of the proposed method. Numerical experiments on a range of elliptic problems demonstrate notable improvements in both prediction accuracy and robustness compared to standard physics-informed neural networks.

Incorporating Local Hölder Regularity into PINNs for Solving Elliptic PDEs

TL;DR

This work tackles solving elliptic PDEs with physics-informed neural networks by embedding local Hölder regularity as a regularization term in the loss. The authors introduce a local Hölder seminorm-based loss, approximated via a variable-distance sampling scheme, and derive a generalization bound linking the augmented loss to the approximation error and quadrature inaccuracies. Under standard stability assumptions, the resulting error estimate shows the solution discrepancy is controlled by the augmented loss and sampling error. Numerical experiments on second-order ODEs, Poisson, variable-coefficient elliptic, and Helmholtz problems demonstrate substantial improvements in accuracy and error distribution compared to standard PINNs, highlighting practical benefits for PDE surrogates and robustness to perturbations.

Abstract

In this paper, local Hölder regularization is incorporated into a physics-informed neural networks (PINNs) framework for solving elliptic partial differential equations (PDEs). Motivated by the interior regularity properties of linear elliptic PDEs, a modified loss function is constructed by introducing local Hölder regularization term. To approximate this term effectively, a variable-distance discrete sampling strategy is developed. Error estimates are established to assess the generalization performance of the proposed method. Numerical experiments on a range of elliptic problems demonstrate notable improvements in both prediction accuracy and robustness compared to standard physics-informed neural networks.

Paper Structure

This paper contains 15 sections, 4 theorems, 52 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak Solution and Interior Regularity
  4. Property of Hölder Seminorm
  5. Methodology
  6. PINNs
  7. Loss Function Guided by Local Hölder Regularity
  8. Calculation Scheme Based on Variable-Distance Sampling
  9. Error Estimation
  10. Numerical Experiment
  11. Second-Order ODE
  12. Poisson Equation
  13. Variable Coefficients Elliptic PDEs
  14. Helmholtz Equation
  15. Summary and Discussion

Key Result

Theorem 2.1

(Interior regularity evans2022partial) Let $k$ be a nonnegative integer, and assume and Suppose furthermore that $u \in H^1(\Omega)$ is a weak solution of the elliptic PDE Then and for each open subset $\Omega^{\prime} \subseteq \Omega$ the estimate holds, where the constant $C$ depending only on $k, \Omega^{\prime}, \Omega$, and the coefficients of $L$.

Figures (7)

  • Figure 1: Network Architecture: The training data includes residual points (upper left figure) and the point set used to calculate the Local Hölder regularization term (lower left figure)
  • Figure 2: The result of Second-Order ODE: $\alpha = 1/2, \rho = 0.01.$
  • Figure 3: The result of Poisson Equation: $N_H=15, \alpha = 1/2, \rho = 0.005.$
  • Figure 4: The result of Variable Coefficients Elliptic PDE: $N_H=20, \alpha = 1/2, \rho = 0.01.$
  • Figure 5: Impact of $N_H$ on Equation (\ref{['quasi-linear']}) ($N_r = 300, \alpha = 1/2, \rho = 0.01$).
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • proof
  • Remark 3.1
  • Theorem 3.1
  • ...and 1 more