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

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 an adaptive collocation point selection strategy utilizing the QR Discrete Empirical Interpolation Method (QR-DEIM), a reduced-order modeling technique for efficiently approximating nonlinear functions. Our results on benchmark PDEs, including the wave, Allen-Cahn, and Burgers' equations, demonstrate that our QR-DEIM-based approach improves PINN accuracy compared to existing methods, offering a promising direction for adaptive collocation point strategies.