Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems

TL;DR

This paper introduces a physics-informed learning framework that efficiently estimates long-term risk probabilities in stochastic systems using limited short-term data, leveraging PDE characterizations for improved generalization and sample efficiency.

Contribution

It develops a novel physics-informed learning approach that combines empirical data with PDE-based risk characterizations to enhance risk estimation in stochastic systems.

Findings

Framework generalizes well beyond training data

Improves sample efficiency and online inference

Provides stable computation of risk probability gradients

Abstract

Accurate estimation of long-term risk is essential for the design and analysis of stochastic dynamical systems. Existing risk quantification methods typically rely on extensive datasets involving risk events observed over extended time horizons, which can be prohibitively expensive to acquire. Motivated by this gap, we propose an efficient method for learning long-term risk probabilities using short-term samples with limited occurrence of risk events. Specifically, we establish that four distinct classes of long-term risk probabilities are characterized by specific partial differential equations (PDEs). Using this characterization, we introduce a physics-informed learning framework that combines empirical data with physics information to infer risk probabilities. We then analyze the theoretical properties of this framework in terms of generalization and convergence. Through numerical experiments, we demonstrate that our framework not only generalizes effectively beyond the sampled states and time horizons but also offers additional benefits such as improved sample efficiency, rapid online inference capabilities under changing system dynamics, and stable computation of probability gradients. These results highlight how embedding PDE constraints, which contain explicit gradient terms and inform how risk probabilities depend on state, time horizon, and system parameters, improves interpolation and generalization between/beyond the available data.