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

Weizheng Zhang1†, Xunjie Xie1†, Hao Pan2, Xiaowei Duan1, Bingteng Sun3, Qiang Du3, Lin Lu1*
1Shandong University, 2Tsinghua University, 3Institute of Engineering Thermophysics, Chinese Academy of Sciences
†Equal Contribution *Corresponding Author
ICML 2026
Paper arXiv
Teaser Image

Abstract

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.

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.

Video

Pipeline

Overview Image

Local Flow Fidelity

We compare velocity magnitude on a representative Gyroid slice at x = 0.5. MUSA-PINN better preserves local flow structures and high-gradient regions near curved boundaries, while baseline PINNs tend to over-smooth the velocity field and produce larger localized errors.

localSlice

Global Mass Conservation

We evaluate physical consistency by tracking the normalized mass flow rate Q(x)/Qin along the streamwise direction. Baseline methods show clear mass leakage inside the domain, whereas MUSA-PINN stays close to the ideal conservation line and preserves flux continuity.

GlobalMassConservation

Longitudinal Flow Development

This longitudinal slice spans the full streamwise direction, x ∈ [0, 5], providing a compact view of how velocity evolves along tortuous passages. Compared with baseline PINNs, MUSA-PINN better preserves the high-velocity structures and remains closer to the CFD reference throughout the channel.

globalSlice

BibTeX

@article{zhang2026musa,
  title={MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries},
  author={Zhang, Weizheng and Xie, Xunjie and Pan, Hao and Duan, Xiaowei and Sun, Bingteng and Du, Qiang and Lu, Lin},
  booktitle={ICML},
  year={2026}
}