Geometric Principles for Machine Learning of Dynamical Systems

2025-02-19

AI Summary

This work addresses the limited structural generalization capability in physical system modeling by proposing a machine learning framework grounded in non-Euclidean geometry—specifically, the manifold of symmetric positive-definite matrices—thereby circumventing manual incorporation of physical priors in model-free architectures. Methodologically, it formalizes generalization as a topological mapping from state space to parameter space, uncovering the geometric essence of symmetry, invariance, and uniqueness; it leverages differential geometry, Lie group/Lie algebra representations, and manifold optimization to achieve geometric parameterization of linear time-invariant (LTI) systems. Contributions include the first rigorous geometric characterization of generalization for dynamical systems and a principled, intrinsic parameterization that respects the underlying physical structure. Experiments demonstrate substantial improvements in extrapolation performance and physical consistency, overcoming structural mismatch inherent in conventional Euclidean embeddings.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

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.

Problem

• Leveraging geometric spaces for machine learning
• Achieving structural generalization in physical systems
• Modeling dynamics on symmetric positive definite manifold

Innovation

• Leveraging non-Euclidean geometric spaces
• Structural generalization in machine learning
• Topological mapping for dynamical systems