Geometric Principles for Machine Learning of Dynamical Systems
This work addresses the challenge of modeling physical systems from data for researchers in machine learning and dynamical systems, but it appears incremental as it builds on existing geometric principles without claiming major breakthroughs.
The paper tackles the problem of achieving structural generalization in machine learning models of dynamical systems by leveraging geometric spaces, proposing that generalization depends on symmetry, invariance, and uniqueness as topological mappings, and illustrates this with linear time-invariant systems on the symmetric positive definite manifold.
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.