首次为神经网络求解纳维-斯托克斯方程提供泛化误差上界，揭示其与粘性系数的关系。
Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations
- 通过控制Rademacher复杂度，推导出深度2神经网络的泛化误差上界。
- 上界不依赖网络宽度，且样本复杂度与维度无关。
- 建议使用新型激活函数，适合流体动力学建模研究者参考。
本文首次建立了针对( d+1 )维不可压缩纳维-斯托克斯方程，通过无监督物理信息神经网络（PINN）框架训练深度2神经网络时的泛化误差上界。该结果基于对PINN损失函数的Rademacher复杂度进行控制。对于权重有界的网络类，所推导的泛化上界不显式依赖网络宽度，并以流体的运动粘度和损失正则化参数表征泛化差距。特别地，所得样本复杂度上界与空间维度无关。这些上界提示应采用新型激活函数求解流体动力学问题。我们在解决泰勒-格林涡基准问题的PINN设置中，对建议的激活函数及对应上界进行了实证验证。
原文摘要 · Abstract (English)
This work establishes rigorous first-of-its-kind upper bounds on the generalization error for the method of approximating solutions to the (d+1)-dimensional incompressible Navier-Stokes equations by training depth-2 neural networks trained via the unsupervised Physics-Informed Neural Network (PINN) framework. This is achieved by bounding the Rademacher complexity of the PINN risk. For appropriately weight bounded net classes our derived generalization bounds do not explicitly depend on the network width and our framework characterizes the generalization gap in terms of the fluid's kinematic viscosity and loss regularization parameters. In particular, the resulting sample complexity bounds are dimension-independent. Our generalization bounds suggest using novel activation functions for solving fluid dynamics. We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor-Green vortex benchmark.
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