  The coincidence between polynomial neural networks and matrix Lie maps is
discussed in the article. The matrix form of Lie transform is an approximation
of the general solution of the nonlinear system of ordinary differential
equations. It can be used for solving systems of differential equations more
efficiently than traditional step-by-step numerical methods. Implementation of
the Lie map as a polynomial neural network provides a tool for both simulation
and data-driven identification of dynamical systems. If the differential
equation is provided, training a neural network is unnecessary. The weights of
the network can be directly calculated from the equation. On the other hand,
for data-driven system learning, the weights can be fitted without any
assumptions in view of differential equations. The proposed technique is
discussed in the examples of both ordinary and partial differential equations.
The building of a polynomial neural network that simulates the Van der Pol
oscillator is discussed. For this example, we consider learning the dynamics
from a single solution of the system. We also demonstrate the building of the
neural network that describes the solution of Burgers' equation that is a
fundamental partial differential equation.
