Machine-Precision Prediction of Low-Dimensional Chaotic Systems

TL;DR

This paper demonstrates that, with high-precision arithmetic and polynomial regression, it is possible to learn low-dimensional chaotic systems from noise-free data with near machine precision accuracy, surpassing previous methods.

Contribution

The authors introduce a high-precision polynomial regression approach that achieves near machine precision learning of chaotic systems, outperforming standard numerical solvers and prior methods.

Findings

Achieves prediction accuracy exceeding standard 64-bit solvers.

Valid prediction times of 32 to 105 Lyapunov times for Lorenz-63.

Extends results to complex and higher-dimensional chaotic systems.

Abstract

Low-dimensional chaotic systems such as the Lorenz-63 model are commonly used to benchmark system-agnostic methods for learning dynamics from data. Here we show that learning from noise-free observations in such systems can be achieved up to machine precision: using ordinary least squares regression on high-degree polynomial features with 512-bit arithmetic, our method exceeds the accuracy of standard 64-bit numerical ODE solvers of the true underlying dynamical systems. Depending on the configuration, we obtain valid prediction times of 32 to 105 Lyapunov times for the Lorenz-63 system, dramatically outperforming prior work that reaches 13 Lyapunov times at most. We further validate our results on Thomas' Cyclically Symmetric Attractor, a non-polynomial chaotic system that is considerably more complex than the Lorenz-63 model, and show that similar results extend also to higher dimensions using the spatiotemporally chaotic Lorenz-96 model. Our findings suggest that learning low-dimensional chaotic systems from noise-free data is a solved problem.