用测度论重构动态系统状态，抗噪抗稀疏，突破经典嵌入局限。
Measure-Theoretic Time-Delay Embedding
- 将系统演化视为概率测度的映射，用最优传输理论构建新嵌入框架
- 在洛伦兹-63、海温与风场重建中均实现高精度状态恢复
- 适合处理含噪声、不完整观测的真实世界动态系统分析
著名的Takens嵌入定理为从部分观测重建动态系统全状态提供了理论基础。然而，经典定理假设系统是确定性的且观测无噪声，限制了其在真实场景中的应用。为此，我们提出一种测度论意义上的推广，采用欧拉描述方式，将嵌入重构视为概率测度空间间的前推映射。数学结果依托最优传输领域的最新进展。基于该测度论时间延迟嵌入理论，我们开发了一种计算方法，旨在从带时间滞后的时间序列部分观测中重建系统全状态，并具备对稀疏和噪声数据的鲁棒性。我们在多个数值实验中评估了该方法，涵盖经典的Lorenz-63系统，以及真实的NOAA海表温度重建与ERA5风场重建任务。
原文摘要 · Abstract (English)
The celebrated Takens' embedding theorem provides a theoretical foundation for reconstructing the full state of a dynamical system from partial observations. However, the classical theorem assumes that the underlying system is deterministic and that observations are noise-free, limiting its applicability in real-world scenarios. Motivated by these limitations, we formulate a measure-theoretic generalization that adopts an Eulerian description of the dynamics and recasts the embedding as a pushforward map between spaces of probability measures. Our mathematical results leverage recent advances in optimal transport. Building on the proposed measure-theoretic time-delay embedding theory, we develop a computational procedure that aims to reconstruct the full state of a dynamical system from time-lagged partial observations, engineered with robustness to handle sparse and noisy data. We evaluate our measure-based approach across several numerical examples, ranging from the classic Lorenz-63 system to real-world applications such as NOAA sea surface temperature reconstruction and ERA5 wind field reconstruction.
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