# Geometric Dictionary Learning of Dynamical Systems with Optimal Transport

arXiv: https://arxiv.org/abs/2605.18276 · HTML: https://mlsanity.com/paper/2605.18276/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 63 references that are in the map.
- First posted on arXiv 18 May 2026.

## Builds on (most cited first, 24 of 24)

- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Provable Meta-Learning of Linear Representations](https://mlsanity.com/paper/2002.11684/) (2020)
- [Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees](https://mlsanity.com/paper/1804.09893/) (2018)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023)
- [Random Fourier Features for Operator-Valued Kernels](https://mlsanity.com/paper/1605.02536/) (2016)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learning](https://mlsanity.com/paper/1608.05639/) (2016)
- [Neural Conditional Probability for Uncertainty Quantification](https://mlsanity.com/paper/2407.01171/) (2024)
- [Dictionary Learning and Sparse Coding on Grassmann Manifolds: An Extrinsic Solution](https://mlsanity.com/paper/1310.4891/) (2013)
- [Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach](https://mlsanity.com/paper/1304.4344/) (2013)
- [Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators](https://mlsanity.com/paper/1805.12324/) (2018)
- [Dynamic Mode Decomposition for data-driven analysis and reduced-order modelling of ExB plasmas: I. Extraction of spatiotemporally coherent patterns](https://mlsanity.com/paper/2308.13726/) (2023)
- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025)
- [Optimization on the biorthogonal manifold](https://mlsanity.com/paper/1609.04161/) (2016)
- [Toeplitz Based Spectral Methods for Data-driven Dynamical Systems](https://mlsanity.com/paper/2602.09791/) (2026)

## Abstract (arXiv)

Learning dynamical systems through operator-theoretic representations provides a powerful framework for analyzing complex dynamics, as spectral quantities such as eigenvalues and invariant structures encode characteristic time scales and long-term behavior. However, dynamical operators are typically estimated independently for each system, preventing the discovery of shared structure across related dynamics. To address this limitation, we posit that related dynamical systems lie near a low-dimensional manifold in spectral operator space. Based on this hypothesis, we introduce DOODL (Dynamical OperatOr Dictionary Learning), a framework that learns a dictionary of characteristic spectral dynamics whose combinations approximate this manifold and yield compact, interpretable embeddings of individual systems. Beyond representation learning, DOODL enables fast and interpretable operator estimation from short and partially observed trajectories by constraining the estimation to the learned operator manifold. Experiments on metastable Langevin dynamics and turbulent plasma simulations demonstrate that DOODL scales to highly complex multiscale regimes while capturing characteristic spectral structure governing the dynamics rather than merely fitting trajectories, achieving errors one to two orders of magnitude lower than independent operator estimation methods in challenging low-data regimes.