From Basins to safe sets: a machine learning perspective on chaotic dynamics

TL;DR

How data driven approaches can accelerate classical tasks such as estimating basin characterization metrics, or partial control of transient chaos, while opening new possibilities for scalable and robust interventions in chaotic systems is highlighted.

Abstract

The study of chaos has long relied on computationally intensive methods to quantify unpredictability and design control strategies. Recent advances in machine learning, from convolutional neural networks to transformer architectures, provide new ways to analyze complex phase space structures and enable real time action in chaotic dynamics. In this perspective article, we highlight how data driven approaches can accelerate classical tasks such as estimating basin characterization metrics, or partial control of transient chaos, while opening new possibilities for scalable and robust interventions in chaotic systems. In recent studies, convolutional networks have reproduced classical basin metrics with negligible bias and low computational cost, while transformer based surrogates have computed accurate safety functions within seconds, bypassing the recursive procedures required by traditional methods. We discuss current opportunities, remaining challenges, and future directions at the intersection of nonlinear dynamics and artificial intelligence.

Figures (2)

• Figure 1: Basins of attraction in a two dimensional dynamical system. Each pixel corresponds to an initial condition, and the color denotes its asymptotic attractor. (a) Example with smooth basin boundaries, where nearby initial conditions converge to the same attractor, indicating high predictability and low basin entropy. (b) Example with fractal basin boundaries, where arbitrarily close initial conditions may converge to different attractors, exemplifying sensitive dependence on initial conditions, increased unpredictability and higher basin entropy.
• Figure 2: Safety function and safe sets. (a) Uncontrolled trajectory (green) escaping the region $Q=[0,1]$. (b) Safety function $U_\infty(x)$ (blue) with admissible control bound $u$ (red), the safe set $S(u)=\{x\in Q: U_\infty(x)\le u\}$ is the yellow region below this bound. (c) Controlled trajectory (green), starting from the same initial condition, confined within $S(u)$. (d) Control signal $u_n$ (cyan), bounded by $u$ (red). This qualitative illustration shows how partial control confines chaotic trajectories with minimal interventions.