Metastability---a phenomenon where systems remain trapped in quasi-stable states before abruptly transitioning under rare perturbations---is ubiquitous in physical systems. Although metastability is a widely observed phenomenon, its identification and analysis present significant challenges. To address these challenges, we propose a novel framework for analyzing metastability using Koopman theory. We use a finite set of system trajectories to learn a representation of the dynamics that defines a latent space in which the system evolves linearly, thereby enabling a systematic characterization of metastable behavior through the spectral properties of the linear mapping. Empirical evaluations demonstrate that our approach is capable of anticipating metastable behavior significantly earlier than its actual manifestation, even with 10% of the simulation duration. Moreover, we establish that the dominant eigenvalue of the learned Koopman matrix in the latent space serves as a critical indicator for detecting metastability across both single-server and multi-server configurations.
Figures & tables
Figure 1. Average latency (top) and Average queue length (bottom) for different arrival rates.
Figure 2. Illustrating the learned model performance for trajectories initialized from empty queue (top plot) and full queue (bottom plot).
Figure 3. The largest eigenvalue of the Koopman matrix for different values of arrival rates.
Figure 4. Illustrating the effect of throttling on the average queue length.
Figure 5. Comparison of average latency under LIFO and FIFO scheduling policies.
Figure 6. Comparison of average queue length under LIFO and FIFO scheduling policies.
Figure 7. Illustrating the performance of the learned model when trained on shorter-duration trajectories.
Figure 8. Variation in settling time in the original space as a function of δ (top plot) and in the latent space as a function of δ^ (bottom plot).
Figure 9. Average latency in the two-server system.
Figure 10. Average queue length in the two-server system.
Figure 11. Variation in settling time as a function of δ^ .
Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.
We study the problem of identifying dynamically distinct basins of attraction in high dimensional time-homogeneous Markov processes using only trajectory sampling. This problem is fundamental in the analysis of metastable dynamical systems, where the process rapidly mixes within basins while transitions between basins occur rarely on the timescale of interest, or even when the state space is reducible. Existing approaches typically rely on spatial discretization or spectral analysis of estimated transition operators, which can become unreliable in high dimensional settings or when the underlying basin geometry is highly nonlinear. We propose a discriminative approach to basin identification based on marginal trajectory distribution comparison. We prove a simple risk separation result: if two initial states belong to the same basin, the Bayes-optimal classifier distinguishing their marginal trajectory distributions achieves risk close to 1/2, whereas if they lie in distinct basins, the optimal risk is close to zero. This observation reduces basin detection to a two-sample discrimination problem between marginal trajectory distributions. Motivated by this principle, we develop a neural algorithm that receives a set of candidate basin representatives and iteratively merges them by estimating classification risk with a neural network that approximates the Bayes classifier. We evaluate the method on various metastable systems. These include synthetic systems constructed by embedding low-dimensional dynamics into high dimensional noisy ambient spaces. In these settings, standard spectral and clustering-based methods often fail, while our approach accurately recovers the underlying basin structure. These results display a shortcoming of existing methods and highlight trajectory discrimination as an effective tool for identifying dynamical basins in high dimensional stochastic systems.
In deep learning, approximation theory motivates increasing representation size. We ask whether this benefit extends to dynamics learning through autoregressive prediction. We analyze the learned time evolution through the eigenstructure of Koopman operators, using relative residuals to detect spurious eigenpairs arising even as one-step error falls. For bounded Koopman operators, we show that minimal residuals over learned dictionary spaces converge pointwise to their full-space counterparts as these spaces approximate the observable space in L2. Our hypothesis is that Koopman spectral reliability helps explain how consistently rollout error decreases with increasing dimension. We compare two models of a shared Koopman autoencoder trained alternately for reconstruction and latent evolution, using latent-prediction loss (one-step prediction errors in latent coordinates) or spectral-residual loss (relative residuals of candidate eigenpairs). Across six chaotic systems, both models reduced median windowed rollout error from smallest to largest dimension. The spectral-residual model achieved lower medians than the latent-prediction model for all systems and dimensions, and its median fell by a larger factor in every system. Its median decreased monotonically with dimension in four systems, against one for latent prediction. Against four baseline families, its mean valid prediction times were nearly always longer. At the largest dimension under two-stage training, we compared eigenvalue positions with each learned dictionary's residual contours. Spectral-residual eigenvalues concentrated in low-residual regions, whereas latent-prediction eigenvalues also appeared in high-residual regions, consistent with the hypothesis.
Itsushi Sakata, Yuta Miyauchi, Yoshinobu Kawahara
RIKEN Center for Advanced Intelligence Project, Tokyo, Japan. · Graduate School of Information Science and Technology, The University of Osaka, Suita, Osaka, Japan.
This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.