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<abstract><p>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} $.</p></abstract>

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

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