Predicting dynamical system evolution with residual neural networks

TL;DR

This paper presents a neural network approach using residual architectures to predict the evolution of dynamical systems from solution samples, effectively modeling ODE systems without explicit knowledge of their functions.

Contribution

It introduces a data-driven neural network method with ResNet architecture for long-term prediction of dynamical systems, outperforming existing models in stability and qualitative accuracy.

Findings

Neural networks accurately reproduce main system dynamics.

Predicted solutions remain stable over longer times.

Method outperforms existing models in stability.

Abstract

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.