Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method

TL;DR

This paper introduces two adaptive collocation point strategies for physics-informed neural networks using QR-DEIM, enhancing accuracy by better capturing critical solution regions in PDEs.

Contribution

It proposes novel QR-DEIM-based adaptive collocation methods for PINNs, addressing limitations of existing sampling techniques and improving PDE solution accuracy.

Findings

QR-DEIM strategies outperform fixed sampling methods

Enhanced accuracy on benchmark PDEs

Adaptive methods better capture solution gradients

Abstract

Physics-informed neural networks (PINNs) have gained significant attention for solving forward and inverse problems related to partial differential equations (PDEs). While advancements in loss functions and network architectures have improved PINN accuracy, the impact of collocation point sampling on their performance remains underexplored. Fixed sampling methods, such as uniform random sampling and equispaced grids, can fail to capture critical regions with high solution gradients, limiting their effectiveness for complex PDEs. Adaptive methods, inspired by adaptive mesh refinement from traditional numerical methods, address this by dynamically updating collocation points during training but may overlook residual dynamics between updates, potentially losing valuable information. To overcome this limitation, we propose two adaptive collocation point selection strategies utilizing the QR Discrete Empirical Interpolation Method (QR-DEIM), a reduced-order modeling technique for efficiently approximating nonlinear functions. Our results on benchmark PDEs demonstrate that our QR-DEIM-based approaches improve PINN accuracy compared to existing methods, offering a promising direction for adaptive collocation point strategies.