Technical Summary: Geometric Dictionary Learning of Dynamical Systems with Optimal Transport

1. Problem Statement

The learning of dynamical systems (DS) from data typically relies on operator-theoretic representations, such as the Koopman operator, where spectral quantities (eigenvalues, invariant subspaces) encode time scales and long-term behavior. However, current methods face two primary limitations:

1. Instance-Specific Estimation: Dynamical operators are typically estimated independently for each system. This prevents the discovery of shared structural patterns across related systems, making transfer learning and comparison ill-posed.
2. Data Scarcity and Geometry: Accurate estimation usually requires long, informative trajectories. In practical settings with short, partial, or metastable trajectories, standard estimators become statistically unstable. Furthermore, the space of dynamical operators is non-Euclidean (spectral and operator-theoretic), rendering classical linear dictionary learning methods inapplicable.

Existing approaches either provide expressive operator representations that remain specific to individual instances or enable principled comparison without supporting the learning of shared structure. There is a lack of a unified framework that captures shared operator structure while enabling geometric operations and transfer.

2. Methodology: DOODL

The authors propose DOODL (Dynamical OperatOr Dictionary Learning), a framework based on the hypothesis that related dynamical systems lie near a low-dimensional manifold in spectral operator space.

2.1 Spectral Geometry and Optimal Transport

DOODL operates on the spectral decompositions of operators. For an operator $G$ with spectral decomposition $G = \sum \lambda_i P_i$ (where $\lambda_i$ are eigenvalues and $P_i$ are spectral projectors), the method represents $G$ as a discrete spectral measure $\mu(G)$.
To compare operators, DOODL utilizes the Spectral-Grassmann Optimal Transport (SGOT) divergence. This metric jointly aligns eigenvalues and invariant subspaces (projectors) via an optimal transport plan, defining a geometry on the space of operators that captures both dynamical time scales and coherent structures.

2.2 Dictionary Learning on a Riemannian Manifold

DOODL learns a dictionary of $d$ characteristic spectral dynamics (atoms) $\mathcal{G} = \{G_1, \dots, G_d\}$.

• Manifold Structure: The space of spectral decompositions is characterized as a smooth Riemannian quotient manifold $\mathcal{M} = \mathcal{N} / (\mathbb{C}^*)^r$, where $\mathcal{N}$ is the space of eigenvalues and bi-orthogonal eigenvectors. This structure accounts for the non-uniqueness of eigenvectors under rescaling.
• Reconstruction Model: Unlike classical dictionary learning which uses linear combinations, DOODL employs a nonlinear reconstruction model. Given coefficients $\alpha$ in the simplex $\Delta^{d-1}$, the reconstruction $B_p(\alpha; \mathcal{G})$ performs a linear combination of spectral components (eigenvalues, left/right eigenvectors) followed by a projection onto the manifold of spectral decompositions to restore the bi-orthogonality constraint. This approach balances computational efficiency with geometric fidelity.
• Optimization: The dictionary and coefficients are learned by minimizing the average SGOT divergence between the observed operators and their reconstructions. The optimization is performed via a stochastic inexact block-coordinate strategy using Riemannian gradient descent on the manifold.

2.3 Dictionary-Based Operator Estimation

A key application of DOODL is estimating operators from short trajectories. Instead of estimating an unconstrained high-dimensional operator, the method restricts the search space to the learned operator manifold (the image of the reconstruction model). This acts as a regularizer, reducing the effective dimensionality of the estimation problem.

3. Theoretical Guarantees

The paper provides statistical guarantees for the operator estimation from short trajectories (Theorem 1).

• Oracle Inequality: The excess risk of the DOODL estimator is bounded by a bias term (approximation error of the learned manifold) and a variance term.
• Dimensionality Reduction: The complexity of the variance term is governed by the intrinsic dimension $m$ of the learned operator manifold rather than the ambient dictionary size $d$.
• Mixing: The bounds hold for $\beta$-mixing trajectories, with the effective sample size adjusted for temporal dependence.
• Bias Decay: The dictionary bias vanishes asymptotically as the length of the training trajectories increases, confirming that the learned manifold converges to the true underlying structure.

4. Experimental Results

The framework was evaluated on two distinct domains:

4.1 Metastable Langevin Dynamics

• Setup: A one-dimensional damped Langevin system with a two-well potential, where the potential width $w$ varies.
• Findings:
  • Manifold Discovery: DOODL successfully identified that the family of systems lies on a low-dimensional (approximately 1D) manifold in operator space. The learned atoms aligned smoothly with the variation in the potential width.
  • Short Trajectory Performance: In regimes with truncated trajectories (up to 10k samples), DOODL achieved SGOT errors one to two orders of magnitude lower than independent Reduced Rank Regression (RRR) estimators and linear dictionary learning baselines.
  • Regime Switching: The method enabled the detection of abrupt regime switches by tracking the evolution of embedding coefficients over sliding windows, demonstrating sharp discontinuities at switching times.

4.2 Turbulent Plasma Dynamics

• Setup: High-dimensional simulations of edge plasma turbulence (Tokam2D) governed by parameters controlling density gradients ($\kappa$) and interchange coupling ($g$).
• Pipeline: The method utilized a frozen PDE foundation model (POSEIDON) to extract spatial descriptors, followed by learning temporal coordinates and estimating finite-rank generator representations.
• Findings:
  • Geometric Organization: The SGOT geometry recovered the 2D manifold structure of the plasma regimes corresponding to $(g, \kappa)$ without supervision.
  • Parameter Recovery: Regressing physical parameters from the learned barycentric coordinates yielded high $R^2$ scores ($0.81$ for $g$, $0.97$ for $\kappa$).
  • Early Identification: Constraining estimation to the learned spectral manifold allowed for reliable operator recovery and parameter identification from short trajectory prefixes (up to 3k time steps), significantly outperforming direct unconstrained estimation. The parameter $\kappa$ (dominant spectral signature) was recovered faster than $g$ (distributed spectral signature).

5. Significance and Claims

The paper claims that DOODL offers a principled framework for transferable representation learning in dynamical systems. Its primary significance lies in:

1. Capturing Shared Structure: It moves beyond instance-specific estimation to learn a shared "dictionary" of characteristic spectral dynamics, enabling the comparison and interpolation of related systems.
2. Data Efficiency: By constraining estimation to a learned low-dimensional spectral manifold, DOODL enables fast and interpretable operator estimation from short, partially observed trajectories where classical methods fail.
3. Geometric Fidelity: It respects the intrinsic non-Euclidean geometry of dynamical operators through Riemannian optimization and Optimal Transport, avoiding the pitfalls of naive linear approximations in operator space.
4. Physical Interpretability: The learned embeddings correspond to physical parameters and dynamical regimes, facilitating system identification and regime shift detection in complex, multiscale environments like plasma turbulence.

The authors acknowledge limitations, noting that the optimization problem is non-convex and the learned manifold is static, not adapting online to distribution shifts. Future work is suggested to explore continual and adaptive variants.