用积分守恒重构流体方程，解决复杂几何下PINN的不稳定性问题。
MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries
- 将流体方程转为球形控制体上的积分守恒，避免点式残差的局部偏差。
- 在TPMS结构上误差降低93%，质量守恒精度显著提升。
- 适合处理拓扑复杂的流体模拟，如多孔介质和微流控结构。
尽管物理信息神经网络（PINNs）提供了求解流体偏微分方程的无网格方法，但标准点式残差最小化在三重周期极小曲面（TPMS）等拓扑复杂域中存在收敛病态问题。点式约束的局部性偏差无法通过曲折通道传播全局信息，导致梯度不稳定和守恒性破坏。为此，我们提出多尺度弱形式物理信息神经网络（MUSA-PINN），将纳维-斯托克斯方程约束重新表述为分层球形控制体上的积分守恒律，并通过控制面上的通量平衡残差强制连续性和动量守恒。方法采用三层子域策略——大体积实现长程耦合、骨架感知的中尺度体积沿输运路径对齐、小体积进行局部细化——并结合两阶段训练流程优先保障连续性。在稳态不可压缩流体于TPMS几何中的实验表明，MUSA-PINN优于现有最优基线，相对误差最高降低93%，且有效保持质量守恒。
原文摘要 · Abstract (English)
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.
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