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A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations

Guangtao Zhang, Jiani Lin, Qijia Zhai, Huiyu Yang, Xujun Chen, Xiaoning Zheng, Ieng Tak Leong

TL;DR

This work analyzes a priori error estimates for Physics-Informed Neural Networks (PINNs) applied to the stiff, strongly nonlinear Allen–Cahn and Cahn–Hilliard equations. It introduces Residuals-RAE, a self-adaptive interior-point weighting scheme that stabilizes training by biasing weights toward regions with larger residuals while maintaining boundedness. The authors prove that a two-hidden-layer tanh network can bound the PINN approximation errors in key quantities (solution, temporal derivative, and nonlinear terms) by the training loss and the number of collocation points, yielding convergence rates of $O(\ln^2(N) N^{-k+2})$ for AC and $O(\ln^2(N) N^{-k+4})$ for CH. Numerical experiments in 1D and 2D corroborate the theoretical results and demonstrate the effectiveness of Residuals-RAE in achieving accurate solutions without requiring domain decomposition in time for CH.

Abstract

This paper aims to analyze errors in the implementation of the Physics-Informed Neural Network (PINN) for solving the Allen--Cahn (AC) and Cahn--Hilliard (CH) partial differential equations (PDEs). The accuracy of PINN is still challenged when dealing with strongly non-linear and higher-order time-varying PDEs. To address this issue, we introduce a stable and bounded self-adaptive weighting scheme known as Residuals-RAE, which ensures fair training and effectively captures the solution. By incorporating this new training loss function, we conduct numerical experiments on 1D and 2D AC and CH systems to validate our theoretical findings. Our theoretical analysis demonstrates that feedforward neural networks with two hidden layers and tanh activation function effectively bound the PINN approximation errors for the solution field, temporal derivative, and nonlinear term of the AC and CH equations by the training loss and number of collocation points.

A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations

TL;DR

This work analyzes a priori error estimates for Physics-Informed Neural Networks (PINNs) applied to the stiff, strongly nonlinear Allen–Cahn and Cahn–Hilliard equations. It introduces Residuals-RAE, a self-adaptive interior-point weighting scheme that stabilizes training by biasing weights toward regions with larger residuals while maintaining boundedness. The authors prove that a two-hidden-layer tanh network can bound the PINN approximation errors in key quantities (solution, temporal derivative, and nonlinear terms) by the training loss and the number of collocation points, yielding convergence rates of for AC and for CH. Numerical experiments in 1D and 2D corroborate the theoretical results and demonstrate the effectiveness of Residuals-RAE in achieving accurate solutions without requiring domain decomposition in time for CH.

Abstract

This paper aims to analyze errors in the implementation of the Physics-Informed Neural Network (PINN) for solving the Allen--Cahn (AC) and Cahn--Hilliard (CH) partial differential equations (PDEs). The accuracy of PINN is still challenged when dealing with strongly non-linear and higher-order time-varying PDEs. To address this issue, we introduce a stable and bounded self-adaptive weighting scheme known as Residuals-RAE, which ensures fair training and effectively captures the solution. By incorporating this new training loss function, we conduct numerical experiments on 1D and 2D AC and CH systems to validate our theoretical findings. Our theoretical analysis demonstrates that feedforward neural networks with two hidden layers and tanh activation function effectively bound the PINN approximation errors for the solution field, temporal derivative, and nonlinear term of the AC and CH equations by the training loss and number of collocation points.
Paper Structure (30 sections, 20 theorems, 122 equations, 13 figures, 4 tables)

This paper contains 30 sections, 20 theorems, 122 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminary Study on the Application of Deep Learning for Solving Partial Differential Equations (PDEs)
  3. Basic setup and notation of Generic PDEs
  4. Deep neural networks architecture
  5. Physics-informed neural networks for approximating the PDEs solution
  6. Self-adaptive learning using the Residuals-RAE weighting scheme
  7. Physics Informed Neural Networks for Approximating the Allen--Cahn Equation
  8. Allen--Cahn Equation
  9. Physics Informed Neural Networks
  10. Error Analysis
  11. Bound on the Residuals
  12. Bounds on the Total Approximation Error
  13. Numerical Examples
  14. 1D Allen--Cahn equation
  15. 2D Allen--Cahn equation
  16. ...and 15 more sections

Key Result

Lemma 2.1

(Ryck2022Estimates) Let $d \geq 2, n \geq 2, m \geq 3, \sigma>0, a_i, b_i \in \mathbb{Z}$ with $a_i<b_i$ for $1 \leq i \leq d, D=\prod_{i=1}^d\left[a_i, b_i\right]$ and $f \in H^m(D)$. Then for every $N \in \mathbb{N}$ with $N>5$ there exists a tanh neural network $u_{dnn}$ with two hidden layers, o and where Moreover, the weights of $u_{dnn}$ scale as $O\left(N^\gamma+N \ln N\right)$ with $\gamm

Figures (13)

  • Figure 1: Structure of fully connected deep neural network.
  • Figure 2: Schematic diagram of Physics-informed neural networks (PINNs).
  • Figure 3: Results for solving 1D Allen--Cahn and 1D Cahn--Hilliard equations using vanilla PINN.
  • Figure 4: Results for solving 1D Allen--Cahn equation using Residuals-RAE.$1^{st}$ column: Reference $u$, $2^{nd}$ column: the predicted $u$ from Residuals-RAE, $3^{rd}$ column: absolute pointwise error (relative $L^2$ error: $8.09e-03$ (initial condition $\# 1$), and $6.53e-02$ (initial condition $\# 2$).
  • Figure 5: Results for solving 1D Allen--Cahn equation using Residuals-RAE.
  • ...and 8 more figures

Theorems & Definitions (21)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.1
  • ...and 11 more