Measure-Theoretic Time-Delay Embedding
This work addresses the limitation of classical embedding theorems for real-world scenarios with noisy data, offering a more robust method for dynamical system reconstruction, though it appears incremental as it builds on existing theory with a new mathematical formulation.
The authors tackled the problem of reconstructing the full state of a dynamical system from partial observations by generalizing Takens' embedding theorem to handle noisy and sparse data, developing a measure-theoretic approach that shows robustness in numerical examples like the Lorenz-63 system and real-world applications such as NOAA sea surface temperature and ERA5 wind field reconstruction.
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.