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

2026-03-09

AI Summary

Standard physics-informed neural networks (PINNs) often suffer from convergence difficulties, gradient instability, and violation of mass conservation when applied to complex topological geometries such as triply periodic minimal surfaces (TPMS), primarily due to the locality of pointwise residuals. To address these limitations, this work proposes a multiscale weak-form PINN framework that enforces conservation laws in an integral form over skeleton-aware mesoscale spherical control volumes. By reformulating the continuity and momentum equations as flux-balance residuals and integrating a three-scale subdomain decomposition with a two-stage training strategy, the method effectively mitigates local bias and rigorously preserves global mass conservation. In steady incompressible flow simulations within TPMS structures, the proposed approach achieves up to a 93% reduction in relative error compared to existing methods, demonstrating significantly improved accuracy and robustness.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Web data generation and simulation

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.

Problem

• Physics-Informed Neural Networks
• Complex Geometries
• Fluid Flow
• Conservation Violations
• Convergence Pathologies

Innovation

• Physics-Informed Neural Networks
• Weak-form formulation
• Multi-scale control volumes
• Flux-balance residuals
• Complex geometries