A Mixed-Form PINNS (MF-PINNS) For Solving The Coupled Stokes-Darcy Equations

TL;DR

This paper introduces MF-PINNs, a novel neural network approach that combines multiple physical formulations with adjusted loss weights to accurately solve coupled Stokes-Darcy equations, especially under challenging parameter conditions.

Contribution

The paper proposes MF-PINNs, an enhanced parallel PINNs method that integrates velocity-pressure and stream-vorticity forms with weighted loss functions to improve solution accuracy.

Findings

Successfully improved accuracy of flow and pressure fields across a wide range of parameters.

Enhanced stability and convergence in solving coupled PDEs with extreme physical constants.

Demonstrated potential for turbulent flow simulations.

Abstract

Parallel physical information neural networks (P-PINNs) have been widely used to solve systems with multiple coupled physical fields, such as the coupled Stokes-Darcy equations with Beavers-Joseph-Saffman (BJS) interface conditions. However, excessively high or low physical constants in partial differential equations (PDE) often lead to ill conditioned loss functions and can even cause the failure of training numerical solutions for PINNs. To solve this problem, we develop a new kind of enhanced parallel PINNs, MF-PINNs, in this article. Our MF-PINNs combines the velocity pressure form (VP) with the stream-vorticity form (SV) and add them with adjusted weights to the total loss functions. The results of numerical experiments show our MF-PINNs have successfully improved the accuracy of the streamline fields and the pressure fields when kinematic viscosity and permeability tensor range from 1e-4 to 1e4. Thus, our MF-PINNs hold promise for more chaotic PDE systems involving turbulent flows. Additionally, we also explore the best combination of the activation functions and their periodicity. And we also try to set the initial learning rate and design its decay strategies. The code and data associated with this paper are available at https://github.com/shxshx48716/MF-PINNs.git.