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.

翻译：神经算子已成为学习无限维函数空间之间映射的变革性工具，在求解复杂偏微分方程方面展现出重要应用价值。本文提出一个严谨的数学框架来分析神经算子的行为特性，重点关注其稳定性、收敛性、聚类动力学、普适性及泛化误差。通过建立一系列新定理，我们在Sobolev空间中给出稳定性界，并借助梯度流解释论证函数空间中的聚类现象，从而为神经算子的设计与优化提供理论指导。基于这些理论保证，我们旨在为未来基于神经算子的方法设计提供清晰统一的框架性指引。