Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences

2025-03-13

AI Summary

Addressing the challenge of characterizing nonlinear dynamical systems from sparse, high-dimensional, and noisy observations, this paper introduces the Smooth Prototype Equivalence (SPE) framework. SPE learns a diffeomorphic mapping—from the data space to a simplified prototype system—via normalizing flows, enabling both behavioral classification and invariant structure estimation. Its key innovation lies in modeling prototype dynamical systems and differentiable manifolds as equivalence classes under smooth conjugacy, thereby supporting equation-agnostic identification of limit cycles and fixed points. Furthermore, SPE enables model-free reconstruction of cell-cycle trajectories directly from single-cell gene expression data. Experiments demonstrate that SPE outperforms state-of-the-art methods in oscillatory system classification; robustly recovers invariant sets from minimal, noisy phase-space observations; and successfully resolves cell-cycle dynamics in real single-cell transcriptomic datasets.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Characterizing dynamical systems given limited measurements is a common challenge throughout the physical and biological sciences. However, this task is challenging, especially due to transient variability in systems with equivalent long-term dynamics. We address this by introducing smooth prototype equivalences (SPE), a framework that fits a diffeomorphism using normalizing flows to distinct prototypes - simplified dynamical systems that define equivalence classes of behavior. SPE enables classification by comparing the deformation loss of the observed sparse, high-dimensional measurements to the prototype dynamics. Furthermore, our approach enables estimation of the invariant sets of the observed dynamics through the learned mapping from prototype space to data space. Our method outperforms existing techniques in the classification of oscillatory systems and can efficiently identify invariant structures like limit cycles and fixed points in an equation-free manner, even when only a small, noisy subset of the phase space is observed. Finally, we show how our method can be used for the detection of biological processes like the cell cycle trajectory from high-dimensional single-cell gene expression data.

Problem

• Characterizing dynamical systems with limited measurements
• Classifying systems with transient variability using SPE
• Detecting biological processes from high-dimensional data

Innovation

• Uses smooth prototype equivalences for dynamics classification
• Employs normalizing flows to fit diffeomorphism to prototypes
• Detects invariant sets via prototype-to-data space mapping