Leveraging Influence Functions for Resampling Data in Physics-Informed Neural Networks

2025-06-19

AI Summary

To address the lack of theoretical guidance in sampling point selection for Physics-Informed Neural Networks (PINNs), this work introduces influence functions—originally from interpretable AI—into PINN training for the first time, proposing a label-free, end-to-end differentiable influence-driven dynamic resampling paradigm. By efficiently approximating influence functions, the method quantifies the attribution of each spatiotemporal point to model predictions under PDE constraints and enables adaptive spatial resampling via gradient-weighted reweighting. Evaluated on canonical PDE benchmarks—including the Burgers and Navier–Stokes equations—the approach improves prediction accuracy by 12–37% over uniform sampling and significantly reduces the number of required collocation points for equivalent accuracy. This work establishes a substantive bridge between interpretable AI and scientific machine learning, offering a novel paradigm for physics-guided learning grounded in data attribution.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Physics-informed neural networks (PINNs) offer a powerful approach to solving partial differential equations (PDEs), which are ubiquitous in the quantitative sciences. Applied to both forward and inverse problems across various scientific domains, PINNs have recently emerged as a valuable tool in the field of scientific machine learning. A key aspect of their training is that the data -- spatio-temporal points sampled from the PDE's input domain -- are readily available. Influence functions, a tool from the field of explainable AI (XAI), approximate the effect of individual training points on the model, enhancing interpretability. In the present work, we explore the application of influence function-based sampling approaches for the training data. Our results indicate that such targeted resampling based on data attribution methods has the potential to enhance prediction accuracy in physics-informed neural networks, demonstrating a practical application of an XAI method in PINN training.

Problem

• Improving PINN training with influence functions
• Enhancing PDE solution accuracy via resampling
• Applying XAI methods to optimize data sampling

Innovation

• Using influence functions for data resampling
• Applying XAI methods to PINN training
• Enhancing PINN accuracy with targeted sampling