PINNfluence: Influence Functions for Physics-Informed Neural Networks

TL;DR

This paper introduces the adaptation of influence functions to physics-informed neural networks (PINNs) to improve their interpretability and debugging capabilities in solving partial differential equations.

Contribution

It presents a novel application of influence functions to PINNs, enabling post-hoc validation and insight into the influence of training points on predictions.

Findings

Influence functions can identify influential collocation points in PINNs.

IFs help reveal how different data points affect PINN predictions.

The method enhances interpretability and debugging of PINNs.

Abstract

Recently, physics-informed neural networks (PINNs) have emerged as a flexible and promising application of deep learning to partial differential equations in the physical sciences. While offering strong performance and competitive inference speeds on forward and inverse problems, their black-box nature limits interpretability, particularly regarding alignment with expected physical behavior. In the present work, we explore the application of influence functions (IFs) to validate and debug PINNs post-hoc. Specifically, we apply variations of IF-based indicators to gauge the influence of different types of collocation points on the prediction of PINNs applied to a 2D Navier-Stokes fluid flow problem. Our results demonstrate how IFs can be adapted to PINNs to reveal the potential for further studies. The code is publicly available at https://github.com/aleks-krasowski/PINNfluence.