Multi evolutional deep neural networks (Multi-EDNN)

TL;DR

This paper introduces Multi-EDNN, a set of methods combining coupled and distributed evolutional deep neural networks to efficiently solve complex PDE systems on large domains, reducing computational costs and improving accuracy.

Contribution

The paper presents Multi-EDNN, a novel framework that employs independent networks for each PDE component and domain partitioning, enhancing scalability and efficiency in solving coupled PDEs.

Findings

Successfully applied to linear advection, heat equation, and Navier-Stokes flows.

Achieved accurate solutions with reduced computational resources.

Demonstrated improved scalability over single-network approaches.

Abstract

Evolutional deep neural networks (EDNN) solve partial differential equations (PDEs) by marching the network representation of the solution fields, using the governing equations. Use of a single network to solve coupled PDEs on large domains requires a large number of network parameters and incurs a significant computational cost. We introduce coupled EDNN (C-EDNN) to solve systems of PDEs by using independent networks for each state variable, which are only coupled through the governing equations. We also introduce distributed EDNN (D-EDNN) by spatially partitioning the global domain into several elements and assigning individual EDNNs to each element to solve the local evolution of the PDE. The networks then exchange the solution and fluxes at their interfaces, similar to flux-reconstruction methods, and ensure that the PDE dynamics are accurately preserved between neighboring elements. Together C-EDNN and D-EDNN form the general class of Multi-EDNN methods. We demonstrate these methods with aid of canonical problems including linear advection, the heat equation, and the compressible Navier-Stokes equations in Couette and Taylor-Green flows.