Artificial neural network as a universal model of nonlinear dynamical systems
This provides an alternative numerical method for solving dynamical ODEs, potentially benefiting researchers in physics and engineering, but it is incremental as it builds on existing neural network approaches.
The authors tackled the problem of modeling nonlinear dynamical systems by proposing a universal artificial neural network map that encodes ODEs directly, without generating numerical time series. They demonstrated high similarity in dynamics for three example systems, including attractors and bifurcation diagrams.
We suggest a universal map capable to recover a behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. Theoretical benefit from this approach is that the universal model admits using common mathematical methods without needing to develop a unique theory for each particular dynamical equations. Form the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Roessler system and also Hindmarch-Rose neuron. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. High similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents.