As an emerging paradigm in scientific machine learning, neural operators aim
to learn operators, via neural networks, that map between infinite-dimensional
function spaces. Several neural operators have been recently developed.
However, all the existing neural operators are only designed to learn operators
defined on a single Banach space, i.e., the input of the operator is a single
function. Here, for the first time, we study the operator regression via neural
networks for multiple-input operators defined on the product of Banach spaces.
We first prove a universal approximation theorem of continuous multiple-input
operators. We also provide detailed theoretical analysis including the
approximation error, which provides a guidance of the design of the network
architecture. Based on our theory and a low-rank approximation, we propose a
novel neural operator, MIONet, to learn multiple-input operators. MIONet
consists of several branch nets for encoding the input functions and a trunk
net for encoding the domain of the output function. We demonstrate that MIONet
can learn solution operators involving systems governed by ordinary and partial
differential equations. In our computational examples, we also show that we can
endow MIONet with prior knowledge of the underlying system, such as linearity
and periodicity, to further improve the accuracy.

本文首次研究了基于神经网络的多输入算子回归问题，并证明了连续多输入算子的通用逼近定理。基于此，结合低秩分解，提出了一个新的神经算子 MIONet，可应用于求解由常微分方程和偏微分方程控制的系统。同时研究了提高神经算子精度的方法，如先验知识的引入。