The core of the demonstration of this paper is to interpret the forward propagation process of machine learning as a parameter estimation problem of nonlinear dynamical systems. This process is to establish a connection between the Recurrent Neural Network and the discrete differential equation, so as to construct a new network structure: ODE-RU. At the same time, under the inspiration of the theory of ordinary differential equations, we propose a new forward propagation mode. In a large number of simulations and experiments, the forward propagation not only shows the trainability of the new architecture, but also achieves a low training error on the basis of main-taining the stability of the network. For the problem requiring long-term memory, we specifically study the obstacle shape reconstruction problem using the backscattering far-field features data set, and demonstrate the effectiveness of the proposed architecture using the data set. The results show that the network can effectively reduce the sensitivity to small changes in the input feature. And the error generated by the ordinary differential equation cyclic unit network in inverting the shape and position of obstacles is less than ${10}^{-2}$.

The primary focus of this research was to investigate the weak Galerkin (WG) finite element method for the Navier-Stokes equations with damping. We established the weak Galerkin finite element numerical scheme and demonstrated the existence and uniqueness of the weak Galerkin numerical solution. Additionally, optimal errors estimates for the velocity and pressure were obtained. Eventually, numerous numerical examples were reported to validate the theoretical analysis.