Polynomial Neural Networks and Taylor maps for Dynamical Systems Simulation and Learning
This provides a method for efficient dynamical systems simulation and learning, applicable when equations are known or unknown, but it is incremental as it builds on existing neural network and numerical integration techniques.
The paper tackles the problem of simulating and learning dynamical systems by connecting polynomial neural networks (PNN) to Taylor maps for solving ordinary differential equations (ODEs), showing that PNN can achieve better accuracy with less computational time compared to traditional numerical solvers.
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.