MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries

While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving fluid-flow PDEs, standard point-wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point-wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi-scale Weak-form PINN (MUSA-PINN), which reformulates Navier-Stokes equation constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux-balance residuals on control surfaces. Our method utilizes a three-scale subdomain strategy-comprising large volumes for long-range coupling, skeleton-aware meso-scale volumes aligned with transport pathways, and small volumes for local refinement-alongside a two-stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA-PINN outperforms state-of-the-art baselines, reducing relative errors by up to 93\% and preserving mass conservation.

Fluid flow through highly complex porous structures is important in many engineering and scientific applications, but it remains difficult to predict accurately. Mesh-free machine-learning solvers are attractive for such geometries, yet many existing methods check the governing physics only at isolated points. In complex domains, this local view can miss how flow and forces are balanced across connected pathways, leading to inaccurate predictions and poor mass conservation. We propose MUSA-PINN, which enforces fluid physics over overlapping regions instead of individual points. These regions act as virtual control volumes that balance fluid and momentum entering and leaving their boundaries. By combining large, pathway-aware, and small regions, MUSA-PINN improves both global consistency and local accuracy, reducing relative errors by up to 93% on complex geometries while better preserving mass conservation.