学习预测混沌：面向混沌动力学鲁棒预测的课程驱动训练
Learning to Predict Chaos: Curriculum-Driven Training for Robust Forecasting of Chaotic Dynamics
AI总结：
针对混沌系统预测中机器学习泛化性差或难学动力学细节的问题，提出基于动力系统复杂度排序的课程训练范式CCF，在多真实基准和架构上显著延长有效预测时长。
摘要：
混沌系统预测是诸多科学领域的核心挑战，哪怕是极微小的预测误差都会被指数级放大，令该问题更趋复杂。现代机器学习方法常陷入两个截然相反的困境：要么过度适配单个已知的混沌系统（如Lorenz-63），导致泛化能力受限；要么不加区分地混合大量无关时间序列，使模型无法学习任何特定动力学机制的细微特征。我们提出课程混沌预测（Curriculum Chaos Forecasting, CCF），这一训练范式可弥合上述差距。CCF基于动力系统理论的基本原理组织训练数据，构建从简单周期行为逐步过渡到高度复杂混沌动力学的课程。我们使用最大李雅普诺夫指数和吸引子维度这两个成熟的混沌度量指标来量化复杂度。通过先在可预测系统上训练序列模型，再逐步引入混沌程度更高的轨迹，CCF能让模型构建出鲁棒且具泛化性的动力学行为表征。我们整理了包含50余个合成常微分方程/偏微分方程（ODE/PDE）系统的库来搭建该课程。实验表明，采用CCF进行预训练能显著提升模型在未见过的真实世界基准上的表现。在太阳黑子数、电力需求、人体心电（ECG）信号等数据集上，与随机顺序训练相比，CCF将有效预测时长提升了40%；与仅使用真实世界数据训练相比，有效预测时长更是提升了一倍以上。我们证实该增益在多种神经架构（GRU、Transformer）上均稳定存在，并通过大量消融实验验证了课程结构的重要性。
英文摘要：
Forecasting chaotic systems is a cornerstone challenge in many scientific fields, complicated by the exponential amplification of even infinitesimal prediction errors. Modern machine learning approaches often falter due to two opposing pitfalls: over-specializing on a single, well-known chaotic system (e.g., Lorenz-63), which limits generalizability, or indiscriminately mixing vast, unrelated time-series, which prevents the model from learning the nuances of any specific dynamical regime. We propose Curriculum Chaos Forecasting (CCF), a training paradigm that bridges this gap. CCF organizes training data based on fundamental principles of dynamical systems theory, creating a curriculum that progresses from simple, periodic behaviors to highly complex, chaotic dynamics. We quantify complexity using the largest Lyapunov exponent and attractor dimension, two well-established metrics of chaos. By first training a sequence model on predictable systems and gradually introducing more chaotic trajectories, CCF enables the model to build a robust and generalizable representation of dynamical behaviors. We curate a library of over 50 synthetic ODE/PDE systems to build this curriculum. Our experiments show that pre-training with CCF significantly enhances performance on unseen, real-world benchmarks. On datasets including Sunspot numbers, electricity demand, and human ECG signals, CCF extends the valid prediction horizon by up to 40% compared to random-order training and more than doubles it compared to training on real-world data alone. We demonstrate that this benefit is consistent across various neural architectures (GRU, Transformer) and provide extensive ablations to validate the importance of the curriculum's structure.
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