This paper focuses on investigating the learning operators for identifying weak solutions to the Navier-Stokes equations. Our objective is to establish a connection between the initial data as input and the weak solution as output. To achieve this, we employ a combination of deep learning methods and compactness argument to derive learning operators for weak solutions for any large initial data in 2D, and for low-dimensional initial data in 3D. Additionally, we utilize the universal approximation theorem to derive a lower bound on the number of sensors required to achieve accurate identification of weak solutions to the Navier-Stokes equations. Our results demonstrate the potential of using deep learning techniques to address challenges in the study of fluid mechanics, particularly in identifying weak solutions to the Navier-Stokes equations.

翻译：本文聚焦于研究识别Navier-Stokes方程弱解的算子。我们的目标是在作为输入的初始数据与作为输出的弱解之间建立联系。为此，我们结合深度学习方法与紧致性论证，推导出适用于二维任意大初始数据及三维低维初始数据的弱解学习算子。此外，我们利用通用逼近定理，推导出实现Navier-Stokes方程弱解准确识别所需传感器数量的下限。结果表明，深度学习方法在应对流体力学研究挑战（尤其是识别Navier-Stokes方程弱解）中具有潜在应用价值。