用傅里叶域符号逼近法，实现对线性微分算子的高效近似。
Approximation Rates in Fréchet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers
- 在弗雷歇度量下，通过傅里叶域符号逼近线性微分算子。
- 给出达到预定误差所需的充分条件，理论严谨。
- 适用于需要高精度算子学习的科学计算场景。
算子学习是近年来利用神经网络模拟偏微分方程（PDE）的一种新方法，其核心思想是学习算子的行为，使训练后的神经网络成为从无限维空间到无限维空间的（近似）映射，从而近似求解由PDE决定的解算子。本文研究了线性微分算子的一般逼近能力，通过在傅里叶域中逼近对应符号来实现。类比霍尔曼-符号的结构，我们考虑由序列半范数诱导的拓扑，并以弗雷歇度量衡量逼近误差。主要结果给出了达到预设逼近误差的充分条件。其次，我们进一步推广主定理，降低了对半范数序列的假设要求。基于指数谱巴伦空间（exponential spectral Barron space）的现有逼近结果，我们给出一个可被良好逼近的具体符号示例。
原文摘要 · Abstract (English)
Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network is an (approximate) mapping in infinite-dimensional spaces that is capable of (approximately) simulating the solution operator governed by the PDE. In our work, we study some general approximation capabilities for linear differential operators by approximating the corresponding symbol in the Fourier domain. Analogous to the structure of the class of Hörmander-Symbols, we consider the approximation with respect to a topology that is induced by a sequence of semi-norms. In that sense, we measure the approximation error in terms of a Fréchet metric, and our main result identifies sufficient conditions for achieving a predefined approximation error. Secondly, we then focus on a natural extension of our main theorem, in which we manage to reduce the assumptions on the sequence of semi-norms. Based on existing approximation results for the exponential spectral Barron space, we then present a concrete example of symbols that can be approximated well.
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