In this work, we propose a method to learn the solution operators of PDEs
defined on varying domains via MIONet, and theoretically justify this method.
We first extend the approximation theory of MIONet to further deal with metric
spaces, establishing that MIONet can approximate mappings with multiple inputs
in metric spaces. Subsequently, we construct a set consisting of some
appropriate regions and provide a metric on this set thus make it a metric
space, which satisfies the approximation condition of MIONet. Building upon the
theoretical foundation, we are able to learn the solution mapping of a PDE with
all the parameters varying, including the parameters of the differential
operator, the right-hand side term, the boundary condition, as well as the
domain. Without loss of generality, we for example perform the experiments for
2-d Poisson equations, where the domains and the right-hand side terms are
varying. The results provide insights into the performance of this method
across convex polygons, polar regions with smooth boundary, and predictions for
different levels of discretization on one task. Reasonably, we point out that
this is a meshless method, hence can be flexibly used as a general solver for a
type of PDE.

提出了一种利用 MIONet 在具有不同域定义的偏微分方程上学习求解算子的方法，并在理论上对此方法进行了验证。通过将 MIONet 的逼近理论扩展到处理度量空间，构造了一组包含适当区域的集合，并在该集合上提供了一种满足 MIONet 逼近条件的度量，从而在处理包括微分算子、右手边项、边界条件和域参数在内的各种参数变化的 PDE 的求解映射方面取得了进展。实验证明该方法在处理凸多边形、具有光滑边界的极坐标区域以及不同离散程度的任务预测方面表现出色。明确地指出，这是一种无网格方法，因此可以作为一种一类 PDE 的通用求解器灵活使用。

通过 MIONet 学习在变动域上定义的偏微分方程的解算符

Learning solution operators of PDEs defined on varying domains via  MIONet

We propose a hybrid iterative method based on MIONet for PDEs, which combines
the traditional numerical iterative solver and the recent powerful machine
learning method of neural operator, and further systematically analyze its
theoretical properties, including the convergence condition, the spectral
behavior, as well as the convergence rate, in terms of the errors of the
discretization and the model inference. We show the theoretical results for the
frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We
give an upper bound of the convergence rate of the hybrid method w.r.t. the
model correction period, which indicates a minimum point to make the hybrid
iteration converge fastest. Several numerical examples including the hybrid
Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are
presented to verify our theoretical results, and also reflect an excellent
acceleration effect. As a meshless acceleration method, it is provided with
enormous potentials for practice applications.

提出了一种基于 MIONet 的混合迭代方法，它结合了传统的数值迭代求解器和最新的神经算子强大机器学习方法，并进一步系统地分析了其理论性能，包括收敛条件、频散行为以及收敛速度，以离散化误差和模型推断的错误为参考，在常用的平滑器（即 Richardson（阻尼 Jacobi）和 Gauss-Seidel）的理论结果中进行了展示。我们给出了混合方法的收敛速度上界，该上界指示了实现混合迭代最快收敛的最佳模型修正周期。通过对 1 维（2 维）泊松方程的混合 Richardson（Gauss-Seidel）迭代的多个数值例子进行验证，证实了我们的理论结果，并且反映出卓越的加速效果。作为一种无网格加速方法，它具有巨大的实际应用潜力。

基于 MIONet 的混合迭代方法在 PDEs 上的理论与数值实例

A hybrid iterative method based on MIONet for PDEs: Theory and numerical  examples

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。