# Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification

arXiv: https://arxiv.org/abs/2412.00589 · HTML: https://mlsanity.com/paper/2412.00589/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 47 references that are in the map.
- First posted on arXiv 30 Nov 2024.

## Builds on (most cited first, 11 of 11)

- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [LUCIE: A Lightweight Uncoupled ClImate Emulator with long-term stability and physical consistency for O(1000)-member ensembles](https://mlsanity.com/paper/2405.16297/) (2024)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures](https://mlsanity.com/paper/2104.15138/) (2021)
- [Deep learning delay coordinate dynamics for chaotic attractors from partial observable data](https://mlsanity.com/paper/2211.11061/) (2022)
- [Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks](https://mlsanity.com/paper/2304.12865/) (2023)
- [Learning Dissipative Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2410.00976/) (2024)
- [Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems](https://mlsanity.com/paper/2101.09411/) (2021)

## Built on by (most cited first, 5 of 5)

- [Transformers for dynamical systems learn transfer operators in-context](https://mlsanity.com/paper/2602.18679/) (2026)
- [The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling](https://mlsanity.com/paper/2605.31547/) (2026)
- [The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery](https://mlsanity.com/paper/2607.18490/) (2026)
- [Goal-oriented learning of stochastic differential equations using error bounds on path-space observables](https://mlsanity.com/paper/2603.20467/) (2026)
- [Conditional Score-Based Modeling of Effective Langevin Dynamics](https://mlsanity.com/paper/2604.23952/) (2026)

## Abstract (arXiv)

While invariant measures are widely employed to analyze physical systems when a direct study of pointwise trajectories is intractable, e.g., due to chaos or noise, they cannot uniquely identify the underlying dynamics. Our first result shows that, in contrast to invariant measures in state coordinates, e.g., [x(t), y(t), z(t)], the invariant measure expressed in time-delay coordinates, e.g., [x(t), x(t-τ),ldots, x(t-(m-1)τ)], can identify the dynamics up to a topological conjugacy. Our second result resolves the remaining ambiguity: by combining invariant measures constructed from multiple delay frames with distinct observables, the system is uniquely identifiable, provided that a suitable initial condition is satisfied. These guarantees require informative observables and appropriate delay parameters (m,τ), which can be limiting in certain settings. We support our theoretical contributions through a series of physical examples demonstrating how invariant measures expressed in delay-coordinates can be used to perform robust system identification in practice.