Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving

Researchers propose a new framework called Operator Learning with Domain Decomposition to solve partial differential equations (PDEs) on arbitrary geometries using neural operators. The approach addresses data efficiency and geometry generalization challenges by breaking complex domains into smaller subdomains that can be solved locally and then combined into global solutions.

• →Neural operators face significant limitations in transferring knowledge to new geometric configurations when solving PDEs.
• →The proposed Schwarz Neural Inference (SNI) scheme partitions complex domains into manageable subdomains for local neural operator solutions.
• →The framework demonstrates improved geometry generalization compared to existing methods across diverse boundary conditions.
• →Theoretical analysis provides convergence rate guarantees and error bounds for the iterative approach.
• →The solution addresses the data-hungry nature of operator learning while maintaining solution accuracy.