标题： 机器精度的低维混沌系统预测 显示英文标题

标题： Machine-Precision Prediction of Low-Dimensional Chaotic Systems

摘要： 低维混沌系统，如Lorenz-63模型，常用于评估从数据中学习动力学的系统无关方法。 在这里，我们展示了在这些系统中从无噪声观测中学习可以达到机器精度：使用高次多项式特征的普通最小二乘回归和512位算术，我们的方法超过了真实底层动力系统的标准64位数值常微分方程求解器的准确性。 根据配置的不同，我们得到Lorenz-63系统的有效预测时间为32到105个李雅普诺夫时间，显著优于之前的工作，最多只能达到13个李雅普诺夫时间。 我们在Thomas' Cyclically Symmetric Attractor上进一步验证了我们的结果，这是一个非多项式的混沌系统，比Lorenz-63模型复杂得多，并且展示了类似的成果也可以通过时空混沌的Lorenz-96模型扩展到更高维度。 我们的发现表明，从无噪声数据中学习低维混沌系统是一个已解决的问题。 显示英文摘要

摘要： 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.