Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems
This work addresses the challenge of expensive data acquisition for risk quantification in safety-critical systems, offering a more efficient method, though it appears incremental as it builds on existing physics-informed approaches.
The paper tackles the problem of estimating long-term risk probabilities in stochastic dynamical systems, which traditionally requires extensive datasets, by proposing a physics-informed learning framework that uses short-term samples and PDE constraints, demonstrating improved sample efficiency and generalization in numerical experiments.
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.