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

TL;DR

This paper introduces a new residuals-weighted loss function for physics-informed neural networks, providing a priori error estimates for solving Allen--Cahn and Cahn--Hilliard equations, supported by theoretical analysis and numerical experiments.

Contribution

It proposes Residuals-RAE, a novel loss function with a pre-training weight update scheme, and derives error bounds for PINNs solving specific PDEs, backed by convergence analysis.

Findings

Residuals-RAE improves PINN accuracy for AC and CH equations.

Theoretical error estimates match numerical experiments.

Neural network architecture bounds approximation errors based on training loss and collocation points.

Abstract

Physics-Informed Neural Networks (PINNs) encounter accuracy limitations when solving the Allen--Cahn (AC) and Cahn--Hilliard (CH) partial differential equations (PDEs). To overcome this, we employ a novel loss function, Residuals-weighted Region Activation Evaluation (Residuals-RAE), featuring a { pre-training weight update scheme}. { Unlike conventional self-adaptive PINNs where weights evolve simultaneously with network parameters, Residuals-RAE-PINNs computes weights from current residuals before each training step and holds them constant during gradient updates. We establish weight convergence under standard neural network optimization assumptions, which justifies analyzing the converged network with constant weights.} Based on this theoretical framework, we derive the error estimation for PINNs with Residuals-RAE when solving AC and CH equations. {The analysis is aligned with Monte-Carlo sampling for the discretization of integrals, consistent with the numerical experiments.} Numerical experiments on one- and two-dimensional AC and CH systems confirm our theoretical results. Additionally, our analysis reveals that feedforward neural networks with two hidden layers and the tanh activation function bound the approximation errors of the solution, its temporal derivative, and the nonlinear term, constrained by the training loss and the number of collocation points.