Neural operator architectures employ neural networks to approximate operators mapping between Banach spaces of functions; they may be used to accelerate model evaluations via emulation, or to discover models from data. Consequently, the methodology has received increasing attention over recent years, giving rise to the rapidly growing field of operator learning. The first contribution of this paper is to prove that for general classes of operators which are characterized only by their $C^r$- or Lipschitz-regularity, operator learning suffers from a curse of dimensionality, defined precisely here in terms of representations of the infinite-dimensional input and output function spaces. The result is applicable to a wide variety of existing neural operators, including PCA-Net, DeepONet and the FNO. The second contribution of the paper is to prove that the general curse of dimensionality can be overcome for solution operators defined by the Hamilton-Jacobi equation; this is achieved by leveraging additional structure in the underlying solution operator, going beyond regularity. To this end, a novel neural operator architecture is introduced, termed HJ-Net, which explicitly takes into account characteristic information of the underlying Hamiltonian system. Error and complexity estimates are derived for HJ-Net which show that this architecture can provably beat the curse of dimensionality related to the infinite-dimensional input and output function spaces.

翻译：神经算子架构利用神经网络近似映射定义在函数巴拿赫空间之间的算子，可用于通过仿真加速模型评估或从数据中发现模型。因此，该方法论近年来越来越受到关注，催生了快速发展的算子学习领域。本文的第一个贡献是证明：对于仅以其$C^r$正则性或Lipschitz正则性为特征的通用算子类，算子学习面临维度灾难（本文基于无限维输入输出函数空间的表示精确定义了这一概念）。该结论适用于多种现有神经算子，包括PCA-Net、DeepONet和FNO。本文的第二个贡献是证明：对于由Hamilton-Jacobi方程定义的解算子，可通过利用底层解算子的附加结构（超越正则性）克服通用维度灾难。为此，本文提出一种新型神经算子架构——HJ-Net，该架构显式纳入底层哈密顿系统的特征信息。针对HJ-Net建立了误差与复杂度估计，表明该架构可证明性地战胜与无限维输入输出函数空间相关的维度灾难。