DD-DeepONet: Domain decomposition and DeepONet for solving partial differential equations in three application scenarios

TL;DR

This paper introduces DD-DeepONet, a scalable framework for efficiently solving PDEs in three practical scenarios involving geometry, boundary conditions, or parameters variations, by decomposing complex geometries and applying transformations.

Contribution

The paper presents DD-DeepONet, a novel domain decomposition approach that enhances DeepONet's scalability and efficiency for solving PDEs with varying geometries, BCs, or parameters.

Findings

Reduces training difficulty and dataset requirements.

Accelerates PDE solution computation.

Demonstrates effectiveness on multiple PDE types.

Abstract

In certain practical engineering applications, there is an urgent need to perform repetitive solving of partial differential equations (PDEs) in a short period. This paper primarily considers three scenarios requiring extensive repetitive simulations. These three scenarios are categorized based on whether the geometry, boundary conditions(BCs), or parameters vary. We introduce the DD-DeepONet, a framework with strong scalability, whose core concept involves decomposing complex geometries into simple structures and vice versa. We primarily study complex geometries composed of rectangles and cuboids, which have numerous practical applications. Simultaneously, stretching transformations are applied to simple geometries to solve shape-dependent problems. This work solves several prototypical PDEs in three scenarios, including Laplace, Poission, N-S, and drift-diffusion equations, demonstrating DD-DeepONet's computational potential. Experimental results demonstrate that DD-DeepONet reduces training difficulty, requires smaller datasets andVRAMper network, and accelerates solution acquisition.