Verifying Physics-Informed Neural Network Fidelity using Classical Fisher Information from Differentiable Dynamical System

TL;DR

This paper introduces a novel framework using Fisher information to rigorously evaluate how well Physics-Informed Neural Networks capture the full dynamical behavior of physical systems, beyond simple trajectory predictions.

Contribution

It proposes a new method employing Fisher information from differentiable dynamical systems to quantify PINN fidelity in representing complex system dynamics and stability.

Findings

Fisher information landscape from PINNs closely matches that of analytical models when trained properly.

The method provides a quantitative measure of PINN accuracy in capturing geometric and stability properties.

Experimental validation using a car dynamical model demonstrates the framework's effectiveness.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving differential equations and modeling physical systems by embedding physical laws into the learning process. However, rigorously quantifying how well a PINN captures the complete dynamical behavior of the system, beyond simple trajectory prediction, remains a challenge. This paper proposes a novel experimental framework to address this by employing Fisher information for differentiable dynamical systems, denoted $g_F^C$. This Fisher information, distinct from its statistical counterpart, measures inherent uncertainties in deterministic systems, such as sensitivity to initial conditions, and is related to the phase space curvature and the net stretching action of the state space evolution. We hypothesize that if a PINN accurately learns the underlying dynamics of a physical system, then the Fisher information landscape derived from the PINN's learned equations of motion will closely match that of the original analytical model. This match would signify that the PINN has achieved comprehensive fidelity capturing not only the state evolution but also crucial geometric and stability properties. We outline an experimental methodology using the dynamical model of a car to compute and compare $g_F^C$ for both the analytical model and a trained PINN. The comparison, based on the Jacobians of the respective system dynamics, provides a quantitative measure of the PINN's fidelity in representing the system's intricate dynamical characteristics.