# Deep learning delay coordinate dynamics for chaotic attractors from partial observable data

arXiv: https://arxiv.org/abs/2211.11061 · HTML: https://mlsanity.com/paper/2211.11061/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 20 Nov 2022.

## Builds on (most cited first, 16 of 16)

- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations](https://mlsanity.com/paper/2108.03561/) (2021)
- [Learning Latent Dynamics for Partially-Observed Chaotic Systems](https://mlsanity.com/paper/1907.02452/) (2019)
- [Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning](https://mlsanity.com/paper/2205.00579/) (2022)
- [Data-driven parameterization of the generalized Langevin equation](https://mlsanity.com/paper/1606.02596/) (2016)
- [Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning](https://mlsanity.com/paper/2202.05750/) (2022)
- [Modeling of Missing Dynamical Systems: Deriving Parametric Models using a Nonparametric Framework](https://mlsanity.com/paper/1905.08082/) (2019)
- [Twisty Takens: A Geometric Characterization of Good Observations on Dense Trajectories](https://mlsanity.com/paper/1809.07131/) (2018)

## Built on by (most cited first, 8 of 8)

- [Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow](https://mlsanity.com/paper/2301.04638/) (2023)
- [Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification](https://mlsanity.com/paper/2412.00589/) (2024)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [Measure-Theoretic Time-Delay Embedding](https://mlsanity.com/paper/2409.08768/) (2024)
- [Characterizing control between interacting subsystems with deep Jacobian estimation](https://mlsanity.com/paper/2507.01946/) (2025)
- [Invariant Measures for Data-Driven Dynamical System Identification: Analysis and Application](https://mlsanity.com/paper/2502.05204/) (2025)
- [Forecasting the Forced van der Pol Equation with Frequent Phase Shifts Using Reservoir Computing](https://mlsanity.com/paper/2404.14651/) (2024)
- [Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics](https://mlsanity.com/paper/2507.21299/) (2025)

## Abstract (arXiv)

A common problem in time series analysis is to predict dynamics with only scalar or partial observations of the underlying dynamical system. For data on a smooth compact manifold, Takens theorem proves a time delayed embedding of the partial state is diffeomorphic to the attractor, although for chaotic and highly nonlinear systems learning these delay coordinate mappings is challenging. We utilize deep artificial neural networks (ANNs) to learn discrete discrete time maps and continuous time flows of the partial state. Given training data for the full state, we also learn a reconstruction map. Thus, predictions of a time series can be made from the current state and several previous observations with embedding parameters determined from time series analysis. The state space for time evolution is of comparable dimension to reduced order manifold models. These are advantages over recurrent neural network models, which require a high dimensional internal state or additional memory terms and hyperparameters. We demonstrate the capacity of deep ANNs to predict chaotic behavior from a scalar observation on a manifold of dimension three via the Lorenz system. We also consider multivariate observations on the Kuramoto-Sivashinsky equation, where the observation dimension required for accurately reproducing dynamics increases with the manifold dimension via the spatial extent of the system.