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. We also show the additional result of the fully-parameterized case in the appendix for interested readers. 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：本文提出了一种通过MIONet学习定义在变化域上的偏微分方程解算子的方法，并从理论上论证了该方法的可行性。我们首先扩展了MIONet的逼近理论，使其进一步适用于度量空间，证明了MIONet能够逼近度量空间中具有多个输入的映射。随后，我们构造了一个由若干合适区域构成的集合，并在此集合上定义度量，从而使其成为满足MIONet逼近条件的度量空间。基于这一理论基础，我们能够学习所有参数（包括微分算子参数、右端项、边界条件以及定义域）均发生变化的偏微分方程的解映射。不失一般性地，我们以二维泊松方程为例开展实验，其中定义域和右端项均发生变化。实验结果揭示了该方法在凸多边形区域、具有光滑边界的极坐标区域，以及同一任务中不同离散化程度的预测性能。对完全参数化情形的补充结果，感兴趣读者可参阅附录。需指出的是，该方法是无网格方法，因此可灵活用作一类偏微分方程的通用求解器。