Matrix Lie Maps and Neural Networks for Solving Differential Equations

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.

Lie Transform Based Polynomial Neural Networks for Dynamical Systems Simulation and Identification

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.