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 rigorously establish a measure-theoretic generalization that adopts an Eulerian description of the dynamics and recasts the embedding as a pushforward map between probability spaces. Our mathematical results leverage recent advances in optimal transportation theory. Building on our novel measure-theoretic time-delay embedding theory, we have developed a new computational framework that forecasts the full state of a dynamical system from time-lagged partial observations, engineered with better robustness to handle sparse and noisy data. We showcase the efficacy and versatility of our approach through several numerical examples, ranging from the classic Lorenz-63 system to large-scale, real-world applications such as NOAA sea surface temperature forecasting and ERA5 wind field reconstruction.

翻译：著名的Takens嵌入定理为从部分观测重构动力系统的完整状态提供了理论基础。然而，经典定理假设底层系统是确定性的且观测无噪声，这限制了其在现实场景中的适用性。受这些局限性的启发，我们严格建立了一个测度论推广，该推广采用动力学的欧拉描述，并将嵌入重述为概率空间之间的前推映射。我们的数学结果利用了最优传输理论的最新进展。基于我们新颖的测度论时滞嵌入理论，我们开发了一个新的计算框架，该框架能够从具有时滞的部分观测中预测动力系统的完整状态，并通过工程设计使其具有更好的鲁棒性以处理稀疏和含噪数据。我们通过多个数值示例展示了我们方法的有效性和通用性，范围从经典的Lorenz-63系统到大规模现实世界应用，如NOAA海面温度预测和ERA5风场重建。