R-PINN: Recovery-type a-posteriori estimator enhanced adaptive PINN

TL;DR

This work addresses the challenge of solving PDEs with large local gradients using physics-informed neural networks (PINNs) by proposing R-PINN, which fuses a recovery-type a-posteriori gradient estimator from adaptive finite element methods with PINN-based adaptivity. It introduces RecAD, an element-wise adaptive distribution strategy that allocates collocation points where the recovered gradient error is largest, and operates in a two-phase training regime to refine solutions efficiently. Across Poisson, Burgers, a two-peak 2D problem, and a wave equation, R-PINN achieves faster convergence with fewer adaptively distributed points and consistently smaller errors than FI-PINN and RBA-PINN, demonstrating a robust hybrid FEM–PINN approach for sharp-gradient PDEs. The method enhances the reliability and efficiency of adaptive PINNs and paves the way for applying recovery-based estimators to broader PDE families.

Abstract

In recent years, with the advancements in machine learning and neural networks, algorithms using physics-informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well-suited for a wide range of equations, they often exhibit suboptimal performance when applied to equations with large local gradients, resulting in substantial localized errors. To address this issue, this paper proposes an adaptive PINN algorithm designed to improve accuracy in such cases. The core idea of the algorithm is to adaptively adjust the distribution of collocation points based on the recovery-type a-posterior error of the current numerical solution, enabling a better approximation of the true solution. This approach is inspired by the adaptive finite element method. By combining the recovery-type a-posteriori estimator, a gradient-recovery estimator commonly used in the adaptive finite element method (FEM) with PINNs, we introduce the Recovery-type a-posteriori estimator enhanced adaptive PINN (R-PINN) and compare its performance with a typical adaptive PINN algorithm, FI-PINN. Our results demonstrate that R-PINN achieves faster convergence with fewer adaptive points and significantly outperforms in the cases with multiple regions of large errors than FI-PINN. Notably, our method is a hybrid numerical approach for solving partial differential equations, integrating adaptive FEM with PINNs.

Figures (20)

• Figure 1: $900$ uniform collocation points generated on $[0, 1]^2$ with three different strategies.
• Figure 2: Flowchart of the R-PINN algorithm.
• Figure 3: Illustration of a $10\times10$ regular diagonal triangulation.
• Figure 4: The distribution of adaptively distributed points in the first $2$ adaptive iterations with two adaptive PINN methods.
• Figure 5: The absolute errors across the domain (left), the cross-sections at $y=0.5$ (middle) and the predicted solutions of three adaptive methods solving Poisson's equation.