# A Mathematical Analysis of Neural Operator Behaviors

arXiv: https://arxiv.org/abs/2410.21481 · HTML: https://mlsanity.com/paper/2410.21481/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 23 references that are in the map.
- First posted on arXiv 28 Oct 2024.

## Builds on (most cited first, 5 of 5)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)

## Built on by (most cited first, 8 of 8)

- [Stochastic Operator Network: A Stochastic Maximum Principle Based Approach to Operator Learning](https://mlsanity.com/paper/2507.10401/) (2025)
- [Split Conformal Prediction in the Function Space with Neural Operators](https://mlsanity.com/paper/2509.04623/) (2025)
- [A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systems](https://mlsanity.com/paper/2603.14863/) (2026)
- [Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operators](https://mlsanity.com/paper/2506.19396/) (2025)
- [ATOM: A Pretrained Neural Operator for Multitask Molecular Dynamics](https://mlsanity.com/paper/2510.05482/) (2025)
- [How Analysis Can Teach Us the Optimal Way to Design Neural Operators](https://mlsanity.com/paper/2411.01763/) (2024)
- [Extending Neural Operators: Robust Handling of Functions Beyond the Training Set](https://mlsanity.com/paper/2603.03621/) (2026)
- [Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation](https://mlsanity.com/paper/2605.08170/) (2026)

## Abstract (arXiv)

Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous mathematical framework for analyzing the behaviors of neural operators, with a focus on their stability, convergence, clustering dynamics, universality, and generalization error. By proposing a list of novel theorems, we provide stability bounds in Sobolev spaces and demonstrate clustering in function space via gradient flow interpretation, guiding neural operator design and optimization. Based on these theoretical gurantees, we aim to offer clear and unified guidance in a single setting for the future design of neural operator-based methods.