Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences

TL;DR

The paper tackles the challenge of identifying invariant sets in nonlinear dynamical systems from sparse measurements. It introduces smooth prototype equivalences (SPE), which learn diffeomorphisms via normalizing flows to map observations to simple prototype dynamics and recover invariant structures in the data space. The approach enables robust classification of oscillatory vs. non-oscillatory dynamics, localization of limit cycles and fixed points, and scalable extension to higher dimensions, including applications to repressilator-like systems and cell-cycle trajectories from single-cell RNA data. This equation-free framework provides a practical tool for uncovering long-term dynamical structures across physics and biology with limited, noisy, high-dimensional data.

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.

Figures (7)

• Figure 1: Schematic for the process of fitting dynamical prototypes using smooth prototype equivalences (SPE). The observed dynamics are compared to each of the chosen prototypes, which involves training a diffeomorphism $H_\theta(x)$, parameterized by a neural network, for each prototype $g(y)$. After training each of the diffeomorphisms, the observed dynamics are classified to the prototype with the smallest equivalence loss. Once the diffeomorphism $H_\theta(x)$ is trained, we have access to a mapping between the data space and the prototype space, which allows us to estimate the long-term behavior, i.e. invariant set, of the dynamics underlying the observed data.
• Figure 2: Estimation of limit cycles using SPE for various 2D dynamical systems.Left: examples of invariant sets predicted from the 500 observed vectors (in gray), for four different dynamical systems exhibiting a limit cycle attractor. For each system, a long ground-truth trajectory was simulated from the hidden system and plotted in black. The red curves are the limit cycles predicted using our method. Right: the average cycle-error of the different systems (simple oscillator (SO), Van der Pol (VDP), Liénard Sigmoid (LS), BZ reaction (BZ), and Selḱov) as a function of the standard deviation of the noise (top) and number of observations (bottom).
• Figure 3: Classification of 2D dynamical systems between node and periodic attractors using SPE. Each point in the scatter represents dynamics from a realization of the different dynamical systems, under a specific choice of parameters defined by the x and y-axis of the plots, with $N=1000$ observed vectors. Each of these was classified by SPE to either a node attractor (red) or periodic attractor (blue). The dashed black line depicts the ground truth bifurcation boundary between periodic and node dynamics. The classification accuracy using SPE, averaged over all parameters considered, is written at the top of each scatter plot.
• Figure 4: Classification accuracy as a function of the number of observed points (left) and amount of observation noise (right). The shaded areas for the different systems represent one standard deviation from the average, which is depicted by the solid line. The black line is the average accuracy over all systems (simple oscillator (SO), BZ reaction, Selḱov, Liénard sigmoid, Liénard polynomial and Van der Pol).
• Figure 5: Classifying dynamics in higher dimensions and recovering the limit cycle from the repressilator system.Top: classification results using SPE on the repressilator system with $N=1000$ observed vectors along a trajectory. Each point in the scatter represents a system with specific parameters that was classified by SPE as either a 2D embedded limit cycle (blue) or node attractor (red). The dashed black line depicts the ground truth bifurcation boundary between periodic and node behavior. Each column depicts dynamics projected onto a different dimensionality, with 2D in the left-most plot and the full 6D system in the right-most plot. Classification accuracy, averaged over all of the parameter settings considered, is written above each of the scatter plots. Bottom: qualitative results of fitting a 6-dimensional repressilator system, projected to the LacI-TetR mRNA plane (left). The gray arrows depict observed position-vector pairs, $(x_i,\dot{x}_i)$, the black line is the ground-truth invariant set of the dynamics and the red line is the predicted invariant set using SPE. Time series of the gene expression (center) and protein levels (right) can be extracted from the predicted limit cycle (shown in red). These are overlayed on top of a trajectory simulated from the underlying (hidden) system, which was simulated for a long time to ensure convergence to the limit cycle, shown in black.