Geometric Principles for Machine Learning of Dynamical Systems

TL;DR

The paper argues that dynamical systems possess rich geometric structure that should guide learning. It proposes a structure-preserving approach on the symmetric (and SPD) matrix manifolds, using $\dot{\mathbf{T}} = A\mathbf{T} + B\mathbf{U}$ with $A=A^T$ and discrete-time propagator $\Phi_A = e^{At}$, enforcing $\Phi_A^T=\Phi_A$ and positive definiteness. A case study on 1D heat conduction shows how Riemannian optimization on the $Sym_n^+$ manifold can jointly identify $\hat{\Phi}_A$ and $\hat{\Phi}_B$ while maintaining geometric constraints, leading to faster convergence and better generalization than model-free baselines. The results suggest that geometry-aware, structure-informed optimization provides stable, interpretable extrapolation across initial conditions and parameter regimes, with potential impact on digital twins and real-time data assimilation.

Abstract

Mathematical descriptions of dynamical systems are deeply rooted in topological spaces defined by non-Euclidean geometry. This paper proposes leveraging structure-rich geometric spaces for machine learning to achieve structural generalization when modeling physical systems from data, in contrast to embedding physics bias within model-free architectures. We consider model generalization to be a function of symmetry, invariance and uniqueness, defined as a topological mapping from state space dynamics to the parameter space. We illustrate this view through the machine learning of linear time-invariant dynamical systems, whose dynamics reside on the symmetric positive definite manifold.

Figures (9)

• Figure 1: Phase portrait of the 1D heat conduction dynamics governed by the eigenstructure of $A$.
• Figure 2: Perturbing parameters of a dynamical system traces a geodesic path on the manifold. In this example, varying the material thickness of our case example results in smooth deformations of the phase portrait geometry as a function of a geodesic path on the $Sym_n^+$ matrix manifold.
• Figure 3: Model training and testing on the London dataset.
• Figure 4: Model testing on the unseen Chicago dataset.
• Figure 5: Five-fold cross validated MSE, for a sweep of forest sizes.