# Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures

arXiv: https://arxiv.org/abs/2104.15138 · HTML: https://mlsanity.com/paper/2104.15138/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2021.

## Built on by (most cited first, 10 of 10)

- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [Discovering Conservation Laws using Optimal Transport and Manifold Learning](https://mlsanity.com/paper/2208.14995/) (2022)
- [Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification](https://mlsanity.com/paper/2412.00589/) (2024)
- [Measure-Theoretic Time-Delay Embedding](https://mlsanity.com/paper/2409.08768/) (2024)
- [Parameter Inference of Time Series by Delay Embeddings and Learning Differentiable Operators](https://mlsanity.com/paper/2203.06269/) (2022)
- [Invariant Measures for Data-Driven Dynamical System Identification: Analysis and Application](https://mlsanity.com/paper/2502.05204/) (2025)
- [Equilibrium flow: From Snapshots to Dynamics](https://mlsanity.com/paper/2509.17990/) (2025)
- [Optimal Transport for Correctional Learning](https://mlsanity.com/paper/2304.01701/) (2023)
- [OGF: An Online Gradient Flow Method for Optimizing the Statistical Steady-State Time Averages of Unsteady Turbulent Flows](https://mlsanity.com/paper/2507.05149/) (2025)
- [Goal-oriented learning of stochastic differential equations using error bounds on path-space observables](https://mlsanity.com/paper/2603.20467/) (2026)

## Abstract (arXiv)

We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data is not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of its physical measure. Thus, we fit physical measures by taking the Wasserstein distance from optimal transportation as a misfit function between two probability distributions. In particular, we analyze the regularity of the resulting loss function for general transportation costs and derive gradient formulas. Physical measures are approximated as fixed points of suitable PDE-based Perron--Frobenius operators. Test cases discussed in the paper include common low-dimensional dynamical systems.