Title: Measure-Theoretic Time-Delay Embedding Show Chinese title

Title: 测度论时间延迟嵌入

Abstract: 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. Show Chinese abstract

Abstract: 著名的Takens嵌入定理为从部分观测中重建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，且观测是无噪声的，这限制了其在现实场景中的适用性。 受这些限制的启发，我们严格建立了采用流体描述的动力学测度论推广，并将嵌入重新表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间延迟嵌入理论，我们开发了一个新的计算框架，可以从时间滞后的部分观测中预测动力系统的完整状态，该框架具有更好的鲁棒性，能够处理稀疏和噪声数据。 我们通过几个数值例子展示了我们方法的有效性和多样性，范围从经典的Lorenz-63系统到大规模的实际应用，如NOAA海面温度预测和ERA5风场重建。