Neural Spline Operators for Risk Quantification in Stochastic Systems
This work addresses risk quantification for safety-critical control in stochastic systems, offering a novel method for handling functional variations, though it appears incremental as an extension of physics-informed neural operators.
The authors tackled the problem of quantifying long-term risk probabilities in stochastic systems with complex varying dynamics by introducing Neural Spline Operators (NeSO), a physics-informed neural operator framework that learns mappings from functional system dynamics to risk probabilities, achieving significant online speed-up over existing methods in case studies.
Accurately quantifying long-term risk probabilities in diverse stochastic systems is essential for safety-critical control. However, existing sampling-based and partial differential equation (PDE)-based methods often struggle to handle complex varying dynamics. Physics-informed neural networks learn surrogate mappings for risk probabilities from varying system parameters of fixed and finite dimensions, yet can not account for functional variations in system dynamics. To address these challenges, we introduce physics-informed neural operator (PINO) methods to risk quantification problems, to learn mappings from varying \textit{functional} system dynamics to corresponding risk probabilities. Specifically, we propose Neural Spline Operators (NeSO), a PINO framework that leverages B-spline representations to improve training efficiency and achieve better initial and boundary condition enforcements, which are crucial for accurate risk quantification. We provide theoretical analysis demonstrating the universal approximation capability of NeSO. We also present two case studies, one with varying functional dynamics and another with high-dimensional multi-agent dynamics, to demonstrate the efficacy of NeSO and its significant online speed-up over existing methods. The proposed framework and the accompanying universal approximation theorem are expected to be beneficial for other control or PDE-related problems beyond risk quantification.