A Mixed-Form PINNs (MF-PINNs) for Solving the Coupled Stokes-Darcy Equations

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This paper is entirely focused on computational fluid dynamics and solving coupled PDE systems (Stokes-Darcy equations). It has no direct relevance to quantitative trading, financial modeling, or market analysis. The only tangential connection is the general use of neural networks and optimization techniques, but the domain-specific content (fluid flow in porous media, interface conditions) is unrelated to finance.

Implementation requires: (1) constructing parallel PINNs with domain decomposition for Stokes and Darcy domains, (2) deriving and implementing both VP and SV forms of the equations, (3) designing mixed-form loss functions with appropriate weight coefficients, (4) implementing periodic activation functions (tanh∘sin), (5) configuring Adam + L-BFGS optimizer pipeline with ReduceLROnPlateau, (6) handling BJS interface conditions, (7) pressure field correction for non-uniqueness. The automatic differentiation framework simplifies some aspects, but the multi-objective loss design and weight tuning add complexity.

Code and data are publicly available on GitHub. Detailed parameter settings (Table 4.1), network architecture, optimizer configurations, and learning rate strategies are provided. Analytical solutions are specified. However, the optimal weight selection for MF-PINNs is somewhat empirical and may require tuning for different problems.