提出基于拉普拉斯特征函数的神经算子，高效逼近反应-扩散系统的解映射。
Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
- 利用拉普拉斯谱表示法构建神经算子，通过特征函数建模解空间。
- 理论证明参数复杂度随精度提升呈多项式增长，突破传统算子学习的参数困境。
- 适用于非线性模式形成问题，适合对偏微分方程求解效率有要求的研究者。
神经算子为学习偏微分方程（PDE）解映射提供了框架，实现对复杂系统的高效代理建模。尽管通用近似性结果已较为清晰，针对非线性反应-扩散系统的近似分析仍有限。本文研究神经算子在广义Gierer-Meinhardt反应-扩散系统中的应用，该系统是非线性模式形成的典型模型。主要结果通过利用隐含于PDE中的格林函数的拉普拉斯谱表示，建立了网络深度、宽度与谱秩相关的显式近似误差界。我们证明所需参数复杂度相对于目标精度最多以多项式方式增长，表明基于拉普拉斯特征函数的神经算子架构可缓解通用算子学习中的参数复杂性难题。在Gierer-Meinhardt系统上的数值实验验证了理论结论。
原文摘要 · Abstract (English)
Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.
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