Predicting dynamical system evolution with residual neural networks
This addresses a common challenge in time series prediction for applications where explicit ODE models are unavailable, though it appears incremental as it builds on known neural network architectures.
The paper tackles the problem of forecasting dynamical systems when the underlying ODE is unknown, using ResNet-like neural networks trained on solution samples, and demonstrates that the models reproduce system dynamics qualitatively well and maintain stability longer than existing models.
Forecasting time series and time-dependent data is a common problem in many applications. One typical example is solving ordinary differential equation (ODE) systems $\dot{x}=F(x)$. Oftentimes the right hand side function $F(x)$ is not known explicitly and the ODE system is described by solution samples taken at some time points. Hence, ODE solvers cannot be used. In this paper, a data-driven approach to learning the evolution of dynamical systems is considered. We show how by training neural networks with ResNet-like architecture on the solution samples, models can be developed to predict the ODE system solution further in time. By evaluating the proposed approaches on three test ODE systems, we demonstrate that the neural network models are able to reproduce the main dynamics of the systems qualitatively well. Moreover, the predicted solution remains stable for much longer times than for other currently known models.