Polynomial Neural Network Approximation of Duffing Equation Solution

This article considers the problem of neural network approximating the solution of the Cauchy problem for a higher order ordinary differential equation. Adaptive approximations of this kind are useful for using as mathematical models of dynamic objects in the creation and adaptation of cyber-physical systems. The approximation scheme is based on the Taylor expansion of the solution with a remainder in the Lagrange form. Furtheron, the second derivative of the solution in the remainder expression is sought in the form of the output of a neural network with a radial basis function (RBF). An algorithm this RBF network training for (selecting the optimal parameters) based on solution data at selected points is presented herein. These data simulate measurements that can be obtained by observing the real object. The algorithm’s main idea involves utilizing the clusterisation for the adjustment of nonlinear input parameters (centres and Gaussian rates of decrease) and pseudoinverse matrix applying for linear input parameters selection. On the example of solving the Cauchy problem for a second order non-linear differential equation, i.e., the Duffing oscillator, a comparison of various approximation schemes is carried out.