从数学上解析神经算子的稳定性与收敛性，为模型设计提供理论支撑。
A Mathematical Analysis of Neural Operator Behaviors
- 构建基于Sobolev空间的稳定性边界，揭示梯度流下的函数空间聚类机制
- 证明神经算子在函数空间中的收敛性与泛化误差控制能力
- 为复杂微分方程求解的神经算子设计提供统一理论指导
神经算子作为学习无限维函数空间映射的革新工具，在求解复杂偏微分方程（PDEs）方面展现出巨大潜力。本文提出一套严谨的数学框架，系统分析神经算子的稳定性、收敛性、聚类动态、通用性及泛化误差。通过一系列新定理，我们在Sobolev空间中给出了稳定性界，并基于梯度流解释展示了函数空间中的聚类现象，为神经算子的设计与优化提供了理论依据。基于这些理论保证，本文旨在为未来基于神经算子的方法设计提供统一且清晰的指导框架。
原文摘要 · Abstract (English)
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.
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