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

2025-01-13

AI Summary

Standard fixed or coarse-grained adaptive sampling in Physics-Informed Neural Networks (PINNs) often fails to resolve high-gradient regions, leading to loss of critical dynamical information. To address this, we propose a residual-driven dynamic collocation selection strategy grounded in QR decomposition and the Discrete Empirical Interpolation Method (QR-DEIM). This is the first work to integrate QR decomposition with DEIM into the PINN adaptive framework, enabling real-time, low-rank, and information-preserving updates of the collocation point set during residual evolution. Evaluated on benchmark problems—including the wave equation, Allen–Cahn equation, and Burgers equation—our method achieves substantial accuracy improvements: average error reductions of 37% compared to uniform sampling and state-of-the-art adaptive approaches. It effectively overcomes key limitations of conventional strategies, namely update latency and insufficient local resolution.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Physical Information Neural Networks
• Optimization
• Complex Mathematical Equations

Innovation

• QR-DEIM
• Adaptive Sampling
• PINNs Optimization