Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems
Zhuoyuan Wang, Albert Chern, Yorie Nakahira
TL;DR
The paper addresses the challenge of estimating long-horizon risk in stochastic safety-critical systems when long-term data are scarce. It derives that four long-term risk probabilities can be characterized by convection-diffusion PDEs on an augmented state, and embeds these PDE constraints into a physics-informed neural network (PIPE) to learn risk probabilities from short-term data. PIPE couples PDE residuals with empirical data, providing bounded-error guarantees and convergence under mild conditions, while enabling rapid online inference, gradient computation, and generalization to unseen states and parameters. The framework demonstrates superior sample efficiency, robustness to parameter changes, and utility as a surrogate for varying dynamics, with broad applicability to stochastic safe control and risk-aware decision-making.
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.