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纳维-斯托克斯方程弱解的算子。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，推导出在二维空间中适用于任意大初始数据、在三维空间中适用于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精确识别纳维-斯托克斯方程弱解所需传感器数量的下界。研究结果表明，深度学习技术在解决流体力学领域的挑战（尤其是识别纳维-斯托克斯方程弱解）方面具有潜力。