Neural Spline Operators for Risk Quantification in Stochastic Systems

2025-08-27

AI Summary

Accurately and efficiently quantifying long-term risk probabilities remains challenging for complex stochastic systems—particularly high-dimensional multi-agent systems exhibiting functional dynamic variations. To address this, we propose the Neural Spline Operator (NeSO), the first physics-informed neural operator framework explicitly designed for risk quantification. NeSO models system dynamics as functional inputs and enables end-to-end mapping from time-varying dynamics to risk probabilities via B-spline parameterization and embedded PDE-based physical constraints. Compared to conventional Monte Carlo sampling and numerical PDE solvers, NeSO offers universal approximation capability, substantially improving training efficiency and fidelity in satisfying initial/boundary conditions. Crucially, under dynamic variation scenarios, NeSO achieves order-of-magnitude speedup in online computation while ensuring verifiability and generalizability—establishing a novel paradigm for real-time, safety-critical risk assessment in adaptive control systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Adversarial Agents
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

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 extit{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.

Problem

• Quantify long-term risk probabilities in stochastic systems
• Handle complex varying functional system dynamics
• Improve training efficiency and boundary condition enforcement

Innovation

• Neural Spline Operators for functional dynamics mapping
• B-spline representations improve training efficiency
• Enforces boundary conditions for accurate quantification