# Neural Spline Operators for Risk Quantification in Stochastic Systems

## TL;DR

This paper introduces Neural Spline Operators, a physics-informed neural operator framework that efficiently learns risk probabilities in complex stochastic systems with functional dynamics, outperforming existing methods in accuracy and speed.

## Contribution

The paper presents Neural Spline Operators (NeSO), a novel PINO framework using B-splines for risk quantification in systems with functional dynamics, with theoretical guarantees and practical advantages.

## Key findings

- NeSO achieves better initial and boundary condition enforcement.
- NeSO demonstrates significant online speed-up over existing methods.
- Theoretical proof of universal approximation capability of NeSO.

## 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 \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.