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

2025-10-20

AI Summary

When solving the Stokes–Darcy coupled system with physics-informed neural networks (PINNs), severe ill-conditioning of the loss function and training divergence arise due to multi-order-of-magnitude variations in physical parameters (e.g., kinematic viscosity and permeability spanning 10⁻⁴–10⁴). To address this, we propose a mixed-formulation PINN (MF-PINN) that simultaneously encodes the velocity–pressure and streamfunction–vorticity representations. The method employs a weighted multi-form loss function, parallel network architecture, periodic activation functions, and adaptive learning rate decay. Its core innovation lies in leveraging complementary physical formulations to mitigate parameter-scale imbalance, thereby substantially enhancing training stability and generalization. Numerical experiments demonstrate that MF-PINNs achieve markedly improved accuracy in both streamline and pressure fields over standard PINNs under strongly nonlinear, multiscale coupling conditions—validating its robustness and practical engineering applicability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

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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.

Problem

• Solving coupled Stokes-Darcy equations with interface conditions
• Addressing ill-conditioned loss from extreme physical constants
• Improving accuracy for chaotic PDE systems like turbulent flows

Innovation

• Combines velocity-pressure and stream-vorticity forms
• Adds adjusted weights to total loss functions
• Improves accuracy for varying physical parameters