# Neural Spline Operators for Risk Quantification in Stochastic Systems

arXiv: https://arxiv.org/abs/2508.20288 · HTML: https://mlsanity.com/paper/2508.20288/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 27 Aug 2025.

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Seismic wave propagation and inversion with Neural Operators](https://mlsanity.com/paper/2108.05421/) (2021)
- [Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validation](https://mlsanity.com/paper/2309.03447/) (2023)
- [Improving physics-informed DeepONets with hard constraints](https://mlsanity.com/paper/2309.07899/) (2023)
- [Safe Control in the Presence of Stochastic Uncertainties](https://mlsanity.com/paper/2104.01259/) (2021)
- [Differentiable Spline Approximations](https://mlsanity.com/paper/2110.01532/) (2021)
- [Myopically Verifiable Probabilistic Certificate for Long-term Safety](https://mlsanity.com/paper/2110.13380/) (2021)
- [A Generalizable Physics-informed Learning Framework for Risk Probability Estimation](https://mlsanity.com/paper/2305.06432/) (2023)
- [Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functions](https://mlsanity.com/paper/2201.00904/) (2022)
- [Building Hybrid B-Spline And Neural Network Operators](https://mlsanity.com/paper/2406.06611/) (2024)
- [Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants](https://mlsanity.com/paper/2412.01232/) (2024)

## Built on by (most cited first, 1 of 1)

- [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026)

## Abstract (arXiv)

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 textitfunctional 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.