Verifying Physics-Informed Neural Network Fidelity using Classical Fisher Information from Differentiable Dynamical System
Josafat Ribeiro Leal Filho, Antônio Augusto Fröhlich
TL;DR
The paper introduces classical Fisher information for deterministic dynamical systems, $g_F^C$, as a geometry-aware fidelity metric to evaluate Physics-Informed Neural Networks (PINNs) beyond trajectory accuracy. It formalizes an experimental framework that compares the Fisher-information landscapes of learned and analytical dynamics, linking fidelity to local stability, curvature, and phase-space deformation via the symmetric Jacobian $F_s$ and its relation to $g_F^C$. Through case studies on kinematic and dynamic bicycle models, the work demonstrates that full physics supervision yields near-perfect FI agreement, while hybrid or inverse-trajectory learning can degrade geometric fidelity and reveal unmodeled forces (e.g., wind, bumps, tire temperature) via FI diagnostics. The approach provides a principled tool for diagnosing missing physics, guiding model augmentation, and increasing trust in PINN-based dynamical predictions with implications for safety-critical control. Overall, the framework advances PINN validation by integrating information geometry with dynamical-system theory to capture stability and sensitivity properties that trajectory-based metrics miss.
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.