2024/09/13 by Jonah Botvinick-Greenhouse, Botvinick-Greenhouse, Jonah, Maria Oprea +5 · 3 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Machine Learning (cs.LG) #Nonlinear Dynamics and Pattern Formation #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2409.08768
openalex publication_date 2024/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.