Title: Lie Transform Based Polynomial Neural Networks for Dynamical Systems Simulation and Identification,
Abstract: In the article, we discuss the architecture of the polynomial neural network
that corresponds to the matrix representation of Lie transform. The matrix form
of Lie transform is an approximation of general solution for the nonlinear
system of ordinary differential equations. Thus, it can be used for simulation
and modeling task. On the other hand, one can identify dynamical system from
time series data simply by optimization of the coefficient matrices of the Lie
transform. Representation of the approach by polynomial neural networks
integrates the strength of both neural networks and traditional model-based
methods for dynamical systems investigation. We provide a theoretical
explanation of learning dynamical systems from time series for the proposed
method, as well as demonstrate it in several applications. Namely, we show
results of modeling and identification for both well-known systems like
Lotka-Volterra equation and more complicated examples from retail,
biochemistry, and accelerator physics.