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

Weizheng Zhang, Xunjie Xie, Hao Pan, Xiaowei Duan, Bingteng Sun, Qiang Du, Lin Lu

TL;DR

The Multi-scale Weak-form PINN (MUSA-PINN) is proposed, which reformulates PDE constraints as integral conservation laws over hierarchical spherical control volumes and enforce continuity and momentum conservation via flux-balance residuals on control surfaces.

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.

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

TL;DR

The Multi-scale Weak-form PINN (MUSA-PINN) is proposed, which reformulates PDE constraints as integral conservation laws over hierarchical spherical control volumes and enforce continuity and momentum conservation via flux-balance residuals on control surfaces.

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.
Paper Structure (70 sections, 48 equations, 14 figures, 6 tables)

This paper contains 70 sections, 48 equations, 14 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. PINNs for Navier-Stokes equations.
  4. Strong-form PINNs and Weak-form PINNs.
  5. Multi-scale PINNs.
  6. Methodology
  7. Preliminaries
  8. Problem Setting.
  9. Nondimensional steady Navier–Stokes.
  10. Boundary conditions.
  11. PINN parameterization.
  12. Weak Formulation via Divergence Theorem
  13. Local integration subdomains.
  14. Weak-form continuity constraint.
  15. Weak-form momentum constraint.
  16. ...and 55 more sections

Figures (14)

  • Figure 1: Flow streamlines on an industrial liquid-cooling plate. (a) CFD reference, (b) PINN-PE baseline (degrades in this complex geometry), and (c) MUSA-PINN. MUSA-PINN achieves a $14.91\%$ relative $\ell_2$ error in velocity w.r.t. CFD.
  • Figure 2: MUSA-PINN pipeline. We sample interior and boundary points for strong-form training and place multi-scale control volumes to impose weak conservation via surface fluxes. A coordinate-to-field MLP with positional encoding maps $(x,y,z)\!\mapsto\!(\mathbf{u},p)$ and is trained in two stages: Stage 1 uses $\mathcal{L}_{bc}+\mathcal{L}_{sf}+\mathcal{L}_{wk,c}$, and Stage 2 adds $\mathcal{L}_{wk,m}$, forming $\mathcal{L}_{total}$.
  • Figure 3: Clipped integration subdomain. A sphere $B(\mathbf{c},r)$ (red) is intersected with the fluid domain $\Omega$ (blue) to form $V(\mathbf{c},r)$.
  • Figure 4: Medium-scale subdomains. These subdomains are centered along the skeletons (grey curves) of flow channels and have diameters that cover channel width.
  • Figure 5: Evaluation of Global Mass Conservation. The plot shows the streamwise evolution of normalized mass flow rate $Q(x)/Q_{in}$ along the flow direction in the Gyroid structure. While standard methods exhibit significant mass leakage deviating from the theoretical value (gray line), MUSA-PINN maintains flux continuity, demonstrating the efficacy of volumetric constraints.
  • ...and 9 more figures