用局部网络+全局损失，少数据预测复杂流体的多尺度运动。
Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies
- 分域局部建模+统一全局损失，捕捉远距离依赖。
- 仅需500个点（<0.3%）就收敛到10^-4误差，比现有方法准。
- 无需实测数据即可模拟湍流分离与再附着，适合工程仿真。
流体运动由非线性纳维-斯托克斯方程支配，即使初始条件可预测，仍呈现多尺度动态。预测此类现象在科学机器学习中仍是巨大挑战，尤其体现在收敛速度、数据需求和解精度方面。复杂流体中，远距离边界条件引发的长程空间依赖进一步加剧了难题，通常需要大量监督数据才能获得满意结果。本文提出分域移位物理信息神经网络（DDS-PINN），通过局部网络与统一全局损失结合，在最小监督下解析多尺度相互作用。该方法在多个基准测试中验证：包括多尺度线性微分方程、非线性布格尔斯方程，以及无数据的平板边界层纳维-斯托克斯模拟。最终应用于计算困难的后向台阶流动（BFS）问题：层流态（Re=100）时，模型结果媲美计算流体力学（CFD），无需任何数据，准确预测边界层厚度、分离区和再附着长度；湍流态（Re=10,000）下，仅用500个随机监督点（占总域<0.3%），即实现10^-4量级收敛，优于残差注意力PINN等主流方法。该方法展现出从稀疏实验测量超分辨率重建复杂湍流流场的强大潜力。
原文摘要 · Abstract (English)
Fluid flows are governed by the nonlinear Navier-Stokes equations, which can manifest multiscale dynamics even from predictable initial conditions. Predicting such phenomena remains a formidable challenge in scientific machine learning, particularly regarding convergence speed, data requirements, and solution accuracy. In complex fluid flows, these challenges are exacerbated by long-range spatial dependencies arising from distant boundary conditions, which typically necessitate extensive supervision data to achieve acceptable results. We propose the Domain-Decomposed and Shifted Physics-Informed Neural Network (DDS-PINN), a framework designed to resolve such multiscale interactions with minimal supervision. By utilizing localized networks with a unified global loss, DDS-PINN captures global dependencies while maintaining local precision. The robustness of the approach is demonstrated across a suite of benchmarks, including a multiscale linear differential equation, the nonlinear Burgers' equation, and data-free Navier-Stokes simulations of flat-plate boundary layers. Finally, DDS-PINN is applied to the computationally challenging backward-facing step (BFS) problem; for laminar regimes (Re = 100), the model yields results comparable to computational fluid dynamics (CFD) without the need for any data, accurately predicting boundary layer thickness, separation, and reattachment lengths. For turbulent BFS flow at Re = 10,000, the framework achieves convergence to O(10^-4) using only 500 random supervision points (< 0.3 % of the total domain), outperforming established methods like Residual-based Attention-PINN in accuracy. This approach demonstrates strong potential for the super-resolution of complex turbulent flows from sparse experimental measurements.
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