Universal replication of chaotic characteristics by classical and quantum machine learning

TL;DR

This paper demonstrates that a single machine learning model, including quantum circuits, can universally replicate chaotic dynamics such as bifurcation diagrams and Lyapunov exponents, outperforming classical models in accuracy and stability.

Contribution

It introduces a universal ML approach for replicating chaotic characteristics across parameter ranges, highlighting quantum circuits' advantages over classical models.

Findings

Quantum circuits outperform LSTM in reproducing chaotic dynamics.

A single ML model can capture bifurcation diagrams for various parameters.

Quantum models show higher accuracy and stability, reducing over-fitting.

Abstract

Replicating chaotic characteristics of non-linear dynamics by machine learning (ML) has recently drawn wide attentions. In this work, we propose that a ML model, trained to predict the state one-step-ahead from several latest historic states, can accurately replicate the bifurcation diagram and the Lyapunov exponents of discrete dynamic systems. The characteristics for different values of the hyper-parameters are captured universally by a single ML model, while the previous works considered training the ML model independently by fixing the hyper-parameters to be specific values. Our benchmarks on the one- and two-dimensional Logistic maps show that variational quantum circuit can reproduce the long-term characteristics with higher accuracy than the long short-term memory (a well-recognized classical ML model). Our work reveals an essential difference between the ML for the chaotic characteristics and that for standard tasks, from the perspective of the relation between performance and model complexity. Our results suggest that quantum circuit model exhibits potential advantages on mitigating over-fitting, achieving higher accuracy and stability.