Machine-Precision Prediction of Low-Dimensional Chaotic Systems

2025-07-13

AI Summary

This study addresses the challenge of long-term, high-precision forecasting of low-dimensional chaotic systems under noise-free observations. We propose an ordinary least squares (OLS) modeling approach leveraging high-degree polynomial features and 512-bit arbitrary-precision arithmetic—deliberately avoiding black-box models such as neural networks. By rigorously controlling both numerical error and model bias, our method achieves machine-precision trajectory prediction (≈10⁻⁶⁴) for canonical systems including the Lorenz-63, Thomas, and Lorenz-96 attractors—the first such result to our knowledge. Effective prediction horizons extend to 32–105 Lyapunov times, substantially exceeding the theoretical limit of ~13 Lyapunov times attained by conventional methods. These findings demonstrate that, for suitably simplified real-world chaotic systems, dynamics can be learned and reconstructed nearly analytically, providing critical empirical evidence for the solvability boundary of chaotic modeling.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Achieve machine-precision prediction in low-dimensional chaotic systems
• Exceed accuracy of standard ODE solvers using high-degree polynomials
• Extend results to complex non-polynomial and higher-dimensional systems

Innovation

• Uses high-degree polynomial features regression
• Employs 512-bit arithmetic for precision
• Validates on multiple chaotic systems