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.

论文证明了对于$C^r$-或Lipschitz-正则性唯一确定的一般类算子而言，算子学习存在维度灾难，但是解决Hamilton-Jacobi方程定义的解算子的一般维度灾难问题时可以利用底层解算子的附加结构，引入了一种新的神经算子结构HJ-Net，该结构明确考虑了底层哈密顿系统的特征信息，从而可以打破无限维输入和输出函数空间相关的维度诅咒。

运算符学习中的维数诅咒