基于非重叠型区域分解的随机参数偏微分方程算子学习OA

Deep neural operators have emerged as a promising paradigm for efficiently solving parametric par-tial differential equations(PDEs)with random parameters via data-driven surrogate modeling.However,di-rectly training global neural operators for large-scale,high-dimensional problems usually suffers from prohibi-tive computational costs and limited generalization capabilities.To address these challenges,this paper pro-poses a novel computational framework that couples nonoverlapping domain decomposition methods(DDM)with operator learning.This approach partitions the global computational domain into several local subdo-mains and utilizes local Karhunen-Loève(KL)expansions to effectively reduce the parameter dimensionality.In the training phase,each local neural operator is trained independently to learn the mapping from local pa-rameters and interface conditions to the local solution.In the prediction phase,an optimization algorithm based on interface constraints is further proposed,enabling all local operators to infer the global approxima-tion rapidly and in parallel.Compared with the global neural operator,domain decomposition significantly re-duces the fitting difficulty of surrogate models and achieves higher prediction accuracy at lower computational cost.A numerical example on a two-dimensional stochastic diffusion equation demonstrates that the proposed local operator requires only about 20％of the network parameters to achieve a better and more robust approxi-mation performance,with the relative error reduced by approximately 34％.Furthermore,its localized struc-ture naturally supports parallel computing,providing a new paradigm for the efficient solution of large-scale parameterized PDEs.