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

Paper: arXiv 2510.17508

Authors: Li Shan, Xi Shen

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides rigorous mathematical derivations for decoupling the Stokes-Darcy system into stream-vorticity form to mitigate gradient competition in PINNs. While the empirical validation is focused on physical simulation benchmark cases rather than financial backtests, the methodology is mathematically sound, highly novel, and fully reproducible with open-source code provided.

Research Flowchart

  flowchart TD
    A[Goal: Solve Stokes-Darcy Equations with Extreme Physical Constants 1e-4 to 1e4] --> B[Inputs: Viscosity & Permeability Tensors, BJS Interface Conditions]
    B --> C[Methodology: Develop Mixed-Form PINNs - MF-PINNs]
    C --> D[Combine Velocity-Pressure VP & Stream-Vorticity SV Forms]
    D --> E[Compute Total Loss Function with Adjusted Weights]
    E --> F[Optimize: Test Activation Functions, Periodicity & LR Decay Strategies]
    F --> G[Outcomes: Overcame Ill-Conditioned Loss, Improved Streamline & Pressure Field Accuracy]