# A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems

arXiv: https://arxiv.org/abs/2509.24920 · HTML: https://mlsanity.com/paper/2509.24920/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 72 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 26 of 26)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Benchmarking Multivariate Time Series Classification Algorithms](https://mlsanity.com/paper/2007.13156/) (2020)
- [Similarity of Neural Network Models: A Survey of Functional and Representational Measures](https://mlsanity.com/paper/2305.06329/) (2023)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction](https://mlsanity.com/paper/2004.00574/) (2020)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis](https://mlsanity.com/paper/2306.10168/) (2023)
- [Discrete Wasserstein Barycenters: Optimal Transport for Discrete Data](https://mlsanity.com/paper/1507.07218/) (2015)
- [Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes](https://mlsanity.com/paper/1801.01990/) (2018)
- [FlowBench: A Large Scale Benchmark for Flow Simulation over Complex Geometries](https://mlsanity.com/paper/2409.18032/) (2024)
- [Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations](https://mlsanity.com/paper/1610.06773/) (2016)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Estimating Koopman operators with sketching to provably learn large scale dynamical systems](https://mlsanity.com/paper/2306.04520/) (2023)
- [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023)
- [Identifying Equivalent Training Dynamics](https://mlsanity.com/paper/2302.09160/) (2023)
- [Learning the Infinitesimal Generator of Stochastic Diffusion Processes](https://mlsanity.com/paper/2405.12940/) (2024)
- [Algorithmic (Semi-)Conjugacy via Koopman Operator Theory](https://mlsanity.com/paper/2209.06374/) (2022)
- [Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators](https://mlsanity.com/paper/1805.12324/) (2018)
- [Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators](https://mlsanity.com/paper/2405.00782/) (2024)
- [Koopman-based spectral clustering of directed and time-evolving graphs](https://mlsanity.com/paper/2204.02951/) (2022)
- [Efficient pseudometrics for data-driven comparisons of nonlinear dynamical systems](https://mlsanity.com/paper/2409.18681/) (2024)

## Built on by (most cited first, 2 of 2)

- [Geometric Dictionary Learning of Dynamical Systems with Optimal Transport](https://mlsanity.com/paper/2605.18276/) (2026)
- [Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification](https://mlsanity.com/paper/2609.35429/) (2026)

## Abstract (arXiv)

The geometry of dynamical systems estimated from trajectory data is a major challenge for machine learning applications. Koopman and transfer operators provide a linear representation of nonlinear dynamics through their spectral decomposition, offering a natural framework for comparison. We propose a novel approach representing each system as a distribution of its joint operator eigenvalues and spectral projectors and defining a metric between systems leveraging optimal transport. The proposed metric is invariant to the sampling frequency of trajectories. It is also computationally efficient, supported by finite-sample convergence guarantees, and enables the computation of Fr'echet means, providing interpolation between dynamical systems. Experiments on simulated and real-world datasets show that our approach consistently outperforms standard operator-based distances in machine learning applications, including dimensionality reduction and classification, and provides meaningful interpolation between dynamical systems.