CDNNs: The coupled deep neural networks for coupling of the Stokes and Darcy-Forchheimer problems

TL;DR

This paper introduces CDNNs, a meshfree deep learning approach that efficiently solves coupled PDEs like Stokes and Darcy-Forchheimer problems by embedding physical constraints into neural networks, demonstrating promising numerical results.

Contribution

The paper presents a novel meshfree deep neural network framework that incorporates physical interface conditions and energy conservation for coupled PDEs, with theoretical convergence guarantees.

Findings

Efficient meshfree solution for coupled PDEs

Incorporates physical interface conditions into neural networks

Demonstrates convergence and numerical performance

Abstract

In this article, we present an efficient deep learning method called coupled deep neural networks (CDNNs) for coupled physical problems. Our method compiles the interface conditions of the coupled PDEs into the networks properly and can be served as an efficient alternative to the complex coupled problems. To impose energy conservation constraints, the CDNNs utilize simple fully connected layers and a custom loss function to perform the model training process as well as the physical property of the exact solution. The approach can be beneficial for the following reasons: Firstly, we sampled randomly and only input spatial coordinates without being restricted by the nature of samples. Secondly, our method is meshfree which makes it more efficient than the traditional methods. Finally, our method is parallel and can solve multiple variables independently at the same time. We give the theory to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. Some numerical experiments are performed and discussed to demonstrate the performance of the proposed method.