Polynomial Neural Networks and Taylor maps for Dynamical Systems Simulation and Learning

TL;DR

This paper explores polynomial neural networks and Taylor maps for efficiently solving and learning dynamical systems modeled by ODEs, offering improved accuracy and computational efficiency.

Contribution

It introduces a novel PNN architecture linked to Taylor maps for ODE simulation and learning, with theoretical analysis and practical examples.

Findings

PNN can simulate ODE dynamics with higher accuracy and less computation.

PNN derived from known ODEs can simulate system behavior without training.

Data-driven PNN fitting enables learning unknown system dynamics.

Abstract

The connection of Taylor maps and polynomial neural networks (PNN) to solve ordinary differential equations (ODEs) numerically is considered. Having the system of ODEs, it is possible to calculate weights of PNN that simulates the dynamics of these equations. It is shown that proposed PNN architecture can provide better accuracy with less computational time in comparison with traditional numerical solvers. Moreover, neural network derived from the ODEs can be used for simulation of system dynamics with different initial conditions, but without training procedure. On the other hand, if the equations are unknown, the weights of the PNN can be fitted in a data-driven way. In the paper we describe the connection of PNN with differential equations in a theoretical way along with the examples for both dynamics simulation and learning with data.