Learning solution operators of PDEs defined on varying domains via MIONet
CoRR(2024)
摘要
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.
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