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

TL;DR

MUSA-PINN introduces a multi-scale weak-form approach to improve physics-informed neural networks for fluid flow in complex geometries, enhancing stability and accuracy over traditional point-wise methods.

Contribution

It proposes a hierarchical multi-scale integral formulation with a specialized training schedule to address convergence issues in complex topologies.

Findings

Reduces relative errors by up to 93%.

Improves mass conservation in complex geometries.

Outperforms existing PINN methods on TPMS geometries.

Abstract

While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving 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 PDE 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.