Geometric Principles for Machine Learning of Dynamical Systems

TL;DR

This paper explores the use of geometric spaces rooted in non-Euclidean geometry to improve machine learning models of dynamical systems, emphasizing structural generalization through symmetry and invariance.

Contribution

It introduces a geometric framework for learning dynamical systems that leverages topological structures instead of physics-based biases in models.

Findings

Demonstrates the approach on linear time-invariant systems

Shows the importance of symmetry and invariance in model generalization

Highlights the role of topological mappings in dynamical systems modeling

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.