四川大学学报（自然科学版）2026，Vol.63Issue(4)：813-822,10.DOI:10.19907/j.0490-6756.260122
基于非重叠型区域分解的随机参数偏微分方程算子学习
Nonoverlapping domain decomposition based operator learning for partial differential equations with stochastic parameters
摘要
Abstract
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.关键词
含参偏微分方程/非重叠型区域分解/算子学习/代理模型Key words
parametric partial differential equation/nonoverlapping domain decomposition/operator learn-ing/surrogate model分类
数理科学引用本文复制引用
刘思琪,石晓宇,徐之航,廖奇峰..基于非重叠型区域分解的随机参数偏微分方程算子学习[J].四川大学学报（自然科学版）,2026,63(4):813-822,10.基金项目
国家自然科学基金(12071291) （12071291）
