Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences (2503.10336v1)

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.

Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences

Overview

The research presented introduces a methodological framework termed Smooth Prototype Equivalences (SPE) for characterizing dynamical systems. SPE addresses the complex challenge of deducing the long-term behaviors and invariant structures of dynamical systems from sparse, high-dimensional observational data. This challenge is pervasive across various scientific fields, such as physics and biology, where measuring the complete system dynamics is often infeasible. The crux of the SPE approach lies in leveraging normalizing flows to fit a diffeomorphism that maps observed data onto simplified prototype systems, thereby identifying the system's long-term dynamical equivalences.

Methodology

The core idea behind SPE is to use prototype learning to classify dynamical systems based on their invariant attributes by constructing diffeomorphisms to predefined prototypes—dynamical systems with known mathematical characteristics. This involves:

1. Prototype Dynamics: Identifying and employing normal forms of dynamical systems that act as prototypes for specific classes of behavior, such as oscillations or fixed points.
2. Equivalence Classification: Fitting normalizing flows, parameterized diffeomorphisms, that align the observed system data with the prototype dynamics. The equivalence loss function is minimized to determine which prototype the observed dynamics closely map to.
3. Invariant Set Estimation: Post-classification, the inverse of the learned mapping enables estimation of the invariant structures, such as limit cycles or fixed points, in the original data space from the prototype space.

Results

The effectiveness of SPE is demonstrated across multiple dynamical systems characterized by differing types of invariant behavior:

• The SPE framework outperformed existing state-of-the-art methods in classifying 2-dimensional oscillatory systems under noisy and sparse observation settings.
• The authors revealed that SPE could reconstruct invariant sets from sparse data, showing high accuracy in estimating limit cycles under varied noise conditions and data sparsities.
• In higher-dimensional systems, SPE was successful in identifying and localizing cyclic dynamics, as illustrated through applications involving the repressilator gene regulatory model.

Implications and Future Directions

SPE enhances the robustness and accuracy of invariant set detection in dynamical systems, crucial for fields like gene expression analysis and aerodynamics, where timely intervention is critical. The method's modular nature allows it to be extended to higher dimensions and various dynamic patterns, including chaotic attractors. Future research could expand SPE's capabilities by enabling unsupervised prototype learning directly from data, particularly valuable for systems with unknown governing equations or multiple invariant structures. Furthermore, investigating the integration of SPE with machine learning models for real-time systems monitoring and control could drive advancements in adaptive dynamic systems management. This approach presents significant implications for predictive modeling and control in engineering, climate science, and beyond, exemplifying a practical solution to deriving insights from complex, nonlinear systems with limited observational data.