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

Li Shan, Xi Shen

TL;DR

The paper tackles ill-conditioning and gradient competition in PINNs for coupled Stokes-Darcy flow with Beavers-Joseph-Saffman interfaces. It introduces MF-PINNs, which fuse velocity-pressure (VP) and stream-vorticity (SV) formulations into a mixed-form loss with carefully chosen weights and decoupling, supplemented by Fourier-feature activations and adaptive learning-rate strategies. Numerical experiments demonstrate that MF-PINNs achieve accurate velocity, streamline, and pressure fields across wide ranges of kinematic viscosity $\nu$ and permeability $\kappa$ (from $10^{-4}$ to $10^{4}$), significantly outperforming baseline and some competing PINN variants. The work provides practical guidance on activation choices, loss weighting, and training schedules, and offers code to enable replication and further exploration of multi-physics PINNs for turbulent or interface-rich problems.

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.

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

TL;DR

The paper tackles ill-conditioning and gradient competition in PINNs for coupled Stokes-Darcy flow with Beavers-Joseph-Saffman interfaces. It introduces MF-PINNs, which fuse velocity-pressure (VP) and stream-vorticity (SV) formulations into a mixed-form loss with carefully chosen weights and decoupling, supplemented by Fourier-feature activations and adaptive learning-rate strategies. Numerical experiments demonstrate that MF-PINNs achieve accurate velocity, streamline, and pressure fields across wide ranges of kinematic viscosity and permeability (from to ), significantly outperforming baseline and some competing PINN variants. The work provides practical guidance on activation choices, loss weighting, and training schedules, and offers code to enable replication and further exploration of multi-physics PINNs for turbulent or interface-rich problems.

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.
Paper Structure (20 sections, 3 theorems, 32 equations, 9 figures, 5 tables)

This paper contains 20 sections, 3 theorems, 32 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Physical modeling
  3. Symbol declaration
  4. The velocity-pressure form of the Stokes-Darcy system
  5. The stream-vorticity form of the Stokes-Darcy system
  6. Algorithm framework
  7. Numerical solutions of DNN
  8. Physical information drives optimization
  9. Gradient competition and MF-PINNs
  10. Activation functions with high-frequency features
  11. Optimizer and learning rate decay
  12. Algorithm design
  13. Numerical test
  14. Model parameter
  15. Metrics for error
  16. ...and 5 more sections

Key Result

Theorem 1

The streamline field $\mathbf{\Psi}_s$ and pressure field $p_s$ of steady Stokes equations could be decoupled as the form $\mathcal{L}_{1}(\mathbf{\Psi}_s,\mathbf{f}_s) = 0$ and $\mathcal{L}_{2}(p_s,\mathbf{f}_s) = 0$. The $\mathcal{L}_{1}(\mathbf{\Psi}_s,\mathbf{f}_s) = 0$ is a fourth-order equatio

Figures (9)

  • Figure 2.1: This cartoon overviews the coupled Stokes-Darcy model with the BJS interface conditions. \newlabelSD.pdf
  • Figure 3.1: This cartoon shows how we combine the VP form and the SV form and design the total new loss function with appropriate weights in the Darcy domain. \newlabelgradcomp.pdf
  • Figure 3.2: This picture illustrates the framework of our MF-PINNs for solving coupled Stokes-Darcy Problems. \newlabelNNframe.pdf
  • Figure 4.1: These images (a)-(d) compare the abilities of PINNs and AT-PINNs. The dashed gray line means that we end up using the Adam optimizer and then use the L-BFGS optimizer. By row: PINNs, AT-PINNs; By column: $\mathit{Loss}$ , $err\mathcal{L}_{2}$. \newlabelLoss_and_L2_12
  • Figure 4.2: These images (a)-(d) compare the abilities of MW-PINNs and our MF-PINNs. The dashed grey line means that we end up using the Adam optimizer and then use the L-BFGS optimizer. By row: MW-PINNs, MF-PINNs; By column: $\mathit{Loss}$ , $err\mathcal{L}_{2}$. \newlabelLoss_and_L2_34
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3