Learning Dynamical Systems with the Spectral Exterior Calculus

TL;DR

This work introduces a data-driven framework for learning dynamical systems on compact Riemannian manifolds by leveraging the spectral exterior calculus (SEC). Vector fields are represented intrinsically as linear combinations of frame elements $B_{ij}=\phi_i\nabla\phi_j$, with coefficients learned from data via a kernel-based Laplacian approximation and a stabilized, truncated regression. The authors prove convergence of the SEC-based reconstructions to the true vector field in the large-data, high-resolution limit and derive orbit-error bounds that quantify trajectory accuracy. Numerical experiments on $S^1$ and $\mathbb{T}^2$ demonstrate accurate vector-field recovery and faithful orbit generation, including favorable out-of-sample extensions. The framework provides a principled, geometry-aware alternative to parametric or black-box methods for learning dynamics on nonlinear manifolds, with potential extensions to global dynamical properties and computational efficiency improvements.

Abstract

We present a data-driven framework for learning dynamical systems on compact Riemannian manifolds based on the spectral exterior calculus (SEC). This approach represents vector fields as linear combinations of frame elements constructed using the eigenfunctions of the Laplacian on smooth functions, along with their gradients. Such reconstructed vector fields generate dynamical flows that consistently approximate the true system, while being compatible with the nonlinear geometry of the manifold. The data-driven implementation of this framework utilizes embedded data points and tangent vectors as training data, along with a graph-theoretic approximation of the Laplacian. In this paper, we prove the convergence of the SEC-based reconstruction in the limit of large data. Moreover, we illustrate the approach numerically with applications to dynamical systems on the unit circle and the 2-torus. In these examples, the reconstructed vector fields compare well with the true vector fields, in terms of both pointwise estimates and generation of orbits.

Figures (9)

• Figure 1: Schematic overview of the training and prediction (inference) phases of our numerical procedure. The training phase has three parts: eigendecomposition of the graph Laplacian (1a), computation of SEC coefficients (1b), and solution of the vector field regression problem (1c). Note that parts 1a and 1b use only the spatial locations of the training data on the manifold, and part 1c uses both locations and vector field data. As a result, parts 1a and 1b may be skipped if the vector field is changed but the sampled points on the manifold are not. The boxes show numerical time complexity estimates based on brute-force linear algebra computations. The actual cost may be improved if specialized numerical techniques are used such as low-rank kernel operator approximations in the inference phase.
• Figure 1: Quiver plots of the true ($\vec{V}$; a, c, e) and SEC-approximated vector fields ($\vec{V}^{(\ell,\alpha)}$; b, d, f) on the unit circle embedded in $\mathbb R^2$. (a, b) Circle rotation, $V_1$; (c, d) connecting arc, $V_2$; and (e, f) variable-speed rotation, $V_3$. The panels show tangent vectors on the unit circle in $\mathbb R^2$ for a subset of points in the training dataset.
• Figure 2: As in \ref{['fig:tangvecs1']}, but for quiver plots of the SEC-approximated vector fields $\vec{V}^{(\ell, \alpha)}$ on the box $[-5, 5] \times [-5, 5] \subset \mathbb R^2$. The unit circle $Y = F(\mathcal{M})$ supporting the sampling distribution of the training data is drawn in each panel for reference. The green lines in panel (b) emanating from the origin depict the angular coordinates of the fixed points of $V_2$ from \ref{['eqn:fixedpoints']}. Notice that $\vec{V}^{(\ell,\alpha)}$ consistently extends the pushforward $\vec{V}$ from $Y$ to other locations in $\mathbb R^2$.
• Figure 3: Dynamical trajectories in $\mathbb R^2$ under the true ($y(t)$; a, c, e) and SEC-approximated vector fields ($y^{(\ell,\alpha)}$; b, d, f) of our circle examples, $V_1$ (a, b), $V_2$ (c,d), and $V_3$ (e, f). All trajectories evolve in a counterclockwise direction.
• Figure 4: As in \ref{['fig:odesols1']}, but showing time series of the $x$ and $y$ components of the $y(t)$ and $y^{(\ell,\alpha)}(t)$ trajectories.

Theorems & Definitions (19)

• Definition 3.1
• Theorem 3.2
• Remark 3.3
• Remark 3.4
• Remark 4.1
• Theorem 5.1
• Lemma 5.2
• Proof 1
• Lemma 5.3
• Lemma 5.4