On the approximation of basins of attraction using deep neural networks

TL;DR

This paper proposes a deep neural network-based method to reconstruct and approximate the basins of attraction in multistable dynamical systems using labeled data, with applications demonstrated on the Lorenz system.

Contribution

It introduces a novel approach to classify initial conditions and approximate basin boundaries using deep learning, linking basin complexity with reconstruction quality.

Findings

Effective basin boundary approximation using neural networks

Application demonstrated on Lorenz system in bistable regime

Relationship between basin entropy and reconstruction accuracy

Abstract

The basin of attraction is the set of initial points that will eventually converge to some attracting set. Its knowledge is important in understanding the dynamical behavior of a given dynamical system of interest. In this work, we address the problem of reconstructing the basins of attraction of a multistable system, using only labeled data. To this end, we view this problem as a classification task and use a deep neural network as a classifier for predicting the attractor that corresponds to any given initial condition. Additionally, we provide a method for obtaining an approximation of the basin boundary of the underlying system, using the trained classification model. Finally, we provide evidence relating the complexity of the structure of the basins of attraction with the quality of the obtained reconstructions, via the concept of basin entropy. We demonstrate the application of the proposed method on the Lorenz system in a bistable regime.