Abstract

In recent works, the authors introduced a neural operator: a special type of neural networks that can approximate maps between infinite-dimensional spaces. Using numerical and analytical techniques, we will highlight the peculiarities of the training and evaluation of these operators. In particular, we will show that, for a broad class of neural operators based on integral transforms, a systematic bias is inevitable, owning to aliasing errors. To avoid this bias, we introduce spectral neural operators based on explicit discretization of the domain and the codomain. Although discretization deteriorates the approximation properties, numerical experiments show that the accuracy of spectral neural operators is often superior to the one of neural operators defined on infinite-dimensional Banach spaces.

中文翻译：

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谱神经算子

摘要

在最近的作品中，作者引入了一种神经算子：一种特殊类型的神经网络，可以近似无限维空间之间的映射。使用数值和分析技术，我们将突出这些操作员的培训和评估的特殊性。特别是，我们将证明，对于基于积分变换的一类广泛的神经算子，由于混叠误差，系统偏差是不可避免的。为了避免这种偏差，我们引入了基于域和共域的显式离散化的谱神经算子。尽管离散化会恶化近似特性，但数值实验表明谱神经算子的精度通常优于无限维 Banach 空间上定义的神经算子。