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.

翻译：本文提出一种通过MIONet学习变分域上定义的偏微分方程（PDE）解算子的方法，并从理论上论证了该方法的有效性。我们首先将MIONet的逼近理论拓展至度量空间，证明了MIONet可逼近度量空间中具有多输入的映射。随后构造了一组包含特定区域的集合，并为其赋予度量结构使其成为满足MIONet逼近条件的度量空间。基于该理论基础，我们能够学习所有参数（包括微分算子参数、右端项、边界条件及计算域）均发生变化的PDE解映射。不失一般性地，我们以二维泊松方程为例进行实验，其中计算域与右端项均动态变化。实验结果揭示了该方法在凸多边形区域、光滑边界极坐标区域中的表现，以及针对同一任务不同离散化程度的预测效果。需要指出的是，该方法属于无网格方法，因而可作为一类PDE的通用求解器灵活使用。