Measure-Theoretic Time-Delay Embedding

TL;DR

This work generalizes Takens' time-delay embedding from trajectories to distributions by formulating a measure-theoretic embedding between probability spaces using pushforwards, grounded in optimal transport on $\mathcal{P}_2(M)$. It establishes a main theorem showing that if the classical delay map is an embedding, its pushforward to probability measures is also an embedding, enabling a robust measure-valued reconstruction of full system states. The authors develop a computational procedure that learns the inverse embedding via a measure-theoretic loss (e.g., $\mathcal{L}_{\text{m}}$ using MMD) and cluster-based empirical measures to handle sparse/noisy data. Demonstrations on Lorenz-63 and related chaotic systems, plus NOAA SST and ERA5 wind datasets, show that the measure-based approach yields smoother reconstructions and better generalization under noise and data scarcity, suggesting a practical, scalable alternative to traditional pointwise learning in data-driven dynamics.

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 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.

Figures (9)

• Figure 1: Illustration of the differences between the classical pointwise time-delay embedding (a) and the proposed measure-theoretic time-delay embedding (b).
• Figure 2: Visualizations of the learned full-state reconstruction map with sparse and noisy data for systems discussed in Section \ref{['subsec:noisydata']}. The comparison includes both pointwise and measure-based approaches against the ground truth.
• Figure 3: Performing time-series prediction for the partially observed Lorenz-63 system from sparse and noisy data. Fig. \ref{['fig:pred3']} shows the distribution of short-time prediction errors in delay coordinates and Fig. \ref{['fig:pred1']} shows long trajectories of the learned delay-coordinate dynamics.
• Figure 4: Visual comparison of the pointwise (see \ref{['eq:pointwise']}) and measure-based (see \ref{['eq:distribution']}) approaches to reconstructing the SST dataset at the testing weeks 425, 550, 675, and 800. The left column features the ground truth snapshot, the middle column shows the measure-based reconstruction, and the right column shows the pointwise reconstruction.
• Figure 5: (Left) Spatial distribution of the average MSE for the measure-based reconstruction of the ERA5 dataset, based upon the Gaussian MMD. (Right) Spatial distribution of the average MSE for the pointwise reconstruction of the ERA5 dataset.

Theorems & Definitions (28)

• Definition 1: Embedding
• Theorem 1: Takens' Embedding Theorem
• Definition 2: The pushforward operator gradient_flows
• Definition 3: Absolute continuity of curves and maps gradient_flows
• Definition 4: Metric derivative gradient_flows
• Theorem 2
• Remark 1
• Remark 2
• Remark 3: Non-uniqueness
• Definition 5: Wasserstein tangent space