We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with (extended) physics informed neural networks. We show that the underlying PDE residual can be made arbitrarily small for tanh neural networks with two hidden layers. Moreover, the total error can be estimated in terms of the training error, network size and number of quadrature points. The theory is illustrated with numerical experiments.

翻译：我们证明了使用（扩展）物理信息神经网络逼近不可压缩纳维-斯托克斯方程所产生的误差的严格界值。结果表明，对于含有两个隐藏层的tanh神经网络，偏微分方程残差可被任意缩小。此外，总误差可通过训练误差、网络规模及求积节点数进行估计。该理论通过数值实验加以验证。