Parameter Estimation for Dynamical Systems Using a Deep Neural Network

Abstract

The deep neural network (DNN) was applied for estimating a set of unknown parameters of a dynamical system whose measured data are given for a set of discrete time points. We developed a new vectorized algorithm that takes the number of unknowns (state variables) and number of parameters into consideration. The algorithm, first, trains the network to determine weights and biases. Next, the algorithm solves the systems of algebraic equations to estimate the parameters of the system. If the right hand side function of the system is smooth and the system have equal numbers of unknowns and parameters, the algorithm solves the algebraic equation at the discrete point where absolute error between the neural network solutions and the measured data is minimum. This improves the accuracy and reduces computational time. Several tests were carried out in linear and non-linear dynamical systems. Last, we showed that the DNN approach is more successful in terms of computational time as the number of hidden layers increases.