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

Matrix Lie Maps and Neural Networks for Solving Differential Equations