用扩散映射核回归实现复杂动力系统的长期精准预测
Learning solution operator of dynamical systems with diffusion maps kernel ridge regression
- 基于扩散映射构建数据驱动核函数，隐式捕捉系统几何结构
- 在混沌吸引子与高维流场上均超越现有方法，精度与效率双优
- 适合追求高效、几何感知的动态系统建模研究者
本文提出一种基于动态感知验证策略的简单核岭回归框架，用于复杂动力系统的长期预测。通过采用由扩散映射导出的数据驱动核函数，所提出的扩散映射核岭回归（DM-KRR）方法无需显式重构流形或建模吸引子，即可隐式适应系统不变集的内在几何结构，从而避免传统方法的性能瓶颈。在涵盖光滑流形、混沌吸引子及高维时空流动的多种系统中，DM-KRR 在准确性和数据效率方面持续优于当前最先进的随机特征、神经网络和算子学习方法。结果表明，长期预测能力不仅依赖模型表达力，更关键在于通过动态一致的模型选择尊重数据中编码的几何约束。简洁性、几何感知性与优异的实证表现共同揭示了一条可靠且高效的复杂动力系统学习路径。
原文摘要 · Abstract (English)
In this work, we propose a simple kernel ridge regression (KRR) framework with a dynamic-aware validation strategy for long-term prediction of complex dynamical systems. By employing a data-driven kernel derived from diffusion maps, the proposed Diffusion Maps Kernel Ridge Regression (DM-KRR) method implicitly adapts to the intrinsic geometry of the system's invariant set, without requiring explicit manifold reconstruction or attractor modeling, procedures that often limit predictive performance. Across a broad range of systems, including smooth manifolds, chaotic attractors, and high-dimensional spatiotemporal flows, DM-KRR consistently outperforms state-of-the-art random feature, neural-network and operator-learning methods in both accuracy and data efficiency. These findings underscore that long-term predictive skill depends not only on model expressiveness, but critically on respecting the geometric constraints encoded in the data through dynamically consistent model selection. Together, simplicity, geometry awareness, and strong empirical performance point to a promising path for reliable and efficient learning of complex dynamical systems.
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