用高精度多项式回归，实现混沌系统超长时预测。
Machine-Precision Prediction of Low-Dimensional Chaotic Systems from Noise-Free Data
- 高次多项式+512位精度，直接拟合动力学规律。
- 对洛伦兹-63模型预测达105个李雅普诺夫时间。
- 适合做高精度科学计算与复杂系统建模的研究者。
低维混沌系统如洛伦兹-63模型常被用于评估无模型方法从数据中学习动力学的能力。本研究发现，在无噪声观测下，通过在高次多项式特征上使用普通最小二乘回归与512位算术，可实现机器精度的建模：该无模型方法的精度与基于系统微分方程的标准64位数值积分器相当。对于洛伦兹-63系统，该方法有效预测时间可达36个李雅普诺夫时间，特定配置下甚至达到105个李雅普诺夫时间，显著优于以往最多仅13个李雅普诺夫时间的成果。该方法在更复杂的托马斯循环对称吸引子（非多项式混沌系统）上也得到验证，并扩展至高维时空混沌的洛伦兹-96模型。结果表明，从无噪声数据中预测低维混沌系统的问题已基本解决。
原文摘要 · Abstract (English)
Low-dimensional chaotic systems such as the Lorenz-63 model are commonly used to benchmark system-agnostic methods for learning dynamics from data. This study shows 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, a system-agnostic method is introduced that matches the accuracy of standard 64-bit numerical ODE solvers using the systems' governing equations. For the Lorenz-63 system, the method achieves valid prediction times of 36 Lyapunov times, and even up to 105 Lyapunov times with favorable precision configurations, dramatically outperforming prior work, which reaches 13 Lyapunov times at most. The results are further validated on Thomas' Cyclically Symmetric Attractor, a non-polynomial chaotic system that is considerably more complex than the Lorenz-63 model, and similar results extend to higher dimensions using the spatiotemporally chaotic Lorenz-96 model. These findings suggest that forecasting low-dimensional chaotic systems from noise-free data is effectively a solved problem.
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