Automated Global Analysis of Experimental Dynamics through Low-Dimensional Linear Embeddings

TL;DR

This work introduces a data-driven computational framework to derive low-dimensional linear models for nonlinear dynamical systems directly from raw experimental data, enabling global stability analysis through interpretable linear models that capture the underlying system structure.

Abstract

Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical modeling, nonlinearity, and high dimensionality. In this work, we introduce a data-driven computational framework to derive low-dimensional linear models for nonlinear dynamical systems directly from raw experimental data. This framework enables global stability analysis through interpretable linear models that capture the underlying system structure. Our approach employs time-delay embedding, physics-informed deep autoencoders, and annealing-based regularization to identify novel low-dimensional coordinate representations, unlocking insights across a variety of simulated and previously unstudied experimental dynamical systems. These new coordinate representations enable accurate long-horizon predictions and automatic identification of intricate invariant sets while providing empirical stability guarantees. Our method offers a promising pathway to analyze complex dynamical behaviors across fields such as physics, climate science, and engineering, with broad implications for understanding nonlinear systems in the real world.

Figures (12)

• Figure 1: Fig. 1 $\vert$ Automated global analysis of experimental dynamics. An overview of our framework to automate the global analysis of experimental dynamical systems by learning low-dimensional latent linear embeddings. a, Collect time-series from a dynamical system. b, Choose the model input dimension by selecting an appropriate time-delay using the mutual-information between trajectories in the system. c, Train a deep autoencoder network that constrains the latent space to behave like a linear dynamical system. d, During training, annealing the coefficient of the loss function and the training prediction horizon to ensure model generalization. e, Long-horizon predictions. f, Interpretable eigenfunction discovery. g, Stability analysis with learned Lyapunov functions.
• Figure 2: Fig. 2 $\vert$ Datasets and prediction error. Diagrams detailing the studied dynamical systems and the prediction error as a function of latent dimension. a, A single pendulum model. b, A circuit with nonlinear resistance known as the Van der Pol oscillator. c, A model for how action potential in neurons are initiated and propagated called the Hodgkin-Huxley Model. d, A model which was devised to study weather predictability known as the Lorenz-96 system. e, A particle mass situated in a double well potential called the Duffing oscillator. f, A mass-spring-damper system with two repelling magnets. g, An experimental magnetic pendulum. h, An experimental double pendulum. i-o, Box and whisker plots showing the mean squared prediction error across embedding dimensions for some of the studied systems. Using our learning approach, the prediction error plateaus with a relatively low-dimensional state space. The red box indicates the latent dimension that the system was modelled in.
• Figure 3: Fig. 3 $\vert$ Long-horizon predictions. Predicted trajectories from low-dimensional linear embeddings of nonlinear dynamics. a, Predicted trajectories in latent space for the single pendulum modeled as a 3D linear system. b, The same latent space trajectories for the pendulum, decomposed into separate modes. c and d, Ground truth and predicted trajectories for angular position and velocity after decoding into state space. e, Predicted latent states for the Van der Pol oscillator as a 3D linear system. f, The same latent space trajectories for the Van der Pol oscillator, decomposed into separate modes. g and h, Ground truth and predicted trajectories for the state variables of the Van der Pol oscillator after decoding. i-l, Ground truth and predicted trajectories in state space for the Hodgkin-Huxley model as a 3D linear system. m and n, Future predicted and ground truth states for the periodic Lorenz-96 model with 40 latitude sectors, represented as a 14D linear system.
• Figure 4: (Caption next page.)
• Figure 4: Fig. 4$\vert$ Long-horizon predictions for multi-stable and chaotic systems.a, The six learned latent states for the Duffing oscillator, forecasting over an extended horizon. b and c, Predicted and ground truth trajectories after decoding, including the correctly anticipated resting attractor. d-f, Forecasted latent variables and states for the Duffing oscillator that come to rest in the opposite potential well compared to the previous trajectory. g-i, Forecasted latent variables (in 6D) and states for the magnetic-mass-spring-damper system with asymmetric basins of attraction. h and i, Predicted and ground truth trajectories that come to rest in the smaller attractor. g-i, Predicted latent variables and states for the magnetic-mass-spring-damper system that come to rest in the larger attractor. m-p, Ground truth and predicted trajectories for each attractor of the experimental magnetic pendulum when modeled as a 6D linear system. q-v, Long-horizon predicted and ground truth trajectories for the measured states of the experimental double pendulum. w-z, Four forecasted and ground truth trajectories for the chaotic Lorenz-96 model with 40 states.