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Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis

Yesom Park, Changhoon Song, Myungjoo Kang

TL;DR

The paper shows that PINNs may fail to converge to PDE solutions even as their loss decreases, due to derivative pathology stemming from uncontrolled gradients. To address this, it introduces VS-PINNs, which augment the primary solution with an auxiliary gradient variable $V$ and constrain $Du=V$, reformulating the PDE into a first-order system and splitting the loss into PDE residual, gradient-matching, and boundary terms. The authors prove convergence of VS-PINNs to generalized solutions in $W^{1,p}$ for second-order linear PDEs: (i) convergence of $u$ follows from the convergence of $V$ and the gradient-matching loss, and (ii) the limit satisfies the weak form of the PDE, yielding a generalized solution. This approach offers gradient control, avoids high-order derivatives in the loss, and broadens the function spaces accessible to neural approximators, with potential extensions to nonlinear or higher-order PDEs and more sophisticated optimization schemes.

Abstract

Physics-informed neural networks (PINNs) have recently emerged as effective methods for solving partial differential equations (PDEs) in various problems. Substantial research focuses on the failure modes of PINNs due to their frequent inaccuracies in predictions. However, most are based on the premise that minimizing the loss function to zero causes the network to converge to a solution of the governing PDE. In this study, we prove that PINNs encounter a fundamental issue that the premise is invalid. We also reveal that this issue stems from the inability to regulate the behavior of the derivatives of the predicted solution. Inspired by the \textit{derivative pathology} of PINNs, we propose a \textit{variable splitting} strategy that addresses this issue by parameterizing the gradient of the solution as an auxiliary variable. We demonstrate that using the auxiliary variable eludes derivative pathology by enabling direct monitoring and regulation of the gradient of the predicted solution. Moreover, we prove that the proposed method guarantees convergence to a generalized solution for second-order linear PDEs, indicating its applicability to various problems.

Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis

TL;DR

The paper shows that PINNs may fail to converge to PDE solutions even as their loss decreases, due to derivative pathology stemming from uncontrolled gradients. To address this, it introduces VS-PINNs, which augment the primary solution with an auxiliary gradient variable and constrain , reformulating the PDE into a first-order system and splitting the loss into PDE residual, gradient-matching, and boundary terms. The authors prove convergence of VS-PINNs to generalized solutions in for second-order linear PDEs: (i) convergence of follows from the convergence of and the gradient-matching loss, and (ii) the limit satisfies the weak form of the PDE, yielding a generalized solution. This approach offers gradient control, avoids high-order derivatives in the loss, and broadens the function spaces accessible to neural approximators, with potential extensions to nonlinear or higher-order PDEs and more sophisticated optimization schemes.

Abstract

Physics-informed neural networks (PINNs) have recently emerged as effective methods for solving partial differential equations (PDEs) in various problems. Substantial research focuses on the failure modes of PINNs due to their frequent inaccuracies in predictions. However, most are based on the premise that minimizing the loss function to zero causes the network to converge to a solution of the governing PDE. In this study, we prove that PINNs encounter a fundamental issue that the premise is invalid. We also reveal that this issue stems from the inability to regulate the behavior of the derivatives of the predicted solution. Inspired by the \textit{derivative pathology} of PINNs, we propose a \textit{variable splitting} strategy that addresses this issue by parameterizing the gradient of the solution as an auxiliary variable. We demonstrate that using the auxiliary variable eludes derivative pathology by enabling direct monitoring and regulation of the gradient of the predicted solution. Moreover, we prove that the proposed method guarantees convergence to a generalized solution for second-order linear PDEs, indicating its applicability to various problems.
Paper Structure (12 sections, 4 theorems, 57 equations)

This paper contains 12 sections, 4 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Notation
  5. PDEs and Generalized Solutions
  6. Physics-Informed Neural Networks (PINNs)
  7. Failure Mode of PINNs
  8. Variable Splitting
  9. Method
  10. Convergence Analysis for Variable Splitting
  11. Discussion and Implications
  12. Conclusion

Key Result

Theorem 2

For any $1\le p<\infty$ and $d\in\mathbb{N}$, there exists a domain $\Omega\subset\mathbb{R}^d$, coefficients $a_{ij}, b_i, c\in C^1\left(\Omega,\mathbb{R}\right)$ for $1\le i,j\le d$, and source functions $f\in C^1\left(\Omega,\mathbb{R}\right)$ and $g\in C\left(\partial\Omega,\mathbb{R}\right)$ wi

Theorems & Definitions (8)

  • Definition 1: Generalized Solutions
  • Theorem 2
  • Remark 3: Derivative pathology of PINNs
  • Definition 4: Variable Splitting for PINNs
  • Remark 5
  • Lemma 6
  • Theorem 7
  • Theorem 8