🤖 AI Summary
This work addresses the challenge of reliably identifying basins of attraction in high-dimensional, time-homogeneous Markovian dynamical systems, where conventional methods struggle with geometric complexity or high dimensionality. The study reframes basin identification as a two-sample discrimination problem on trajectory distributions: trajectories originating from the same basin exhibit similar distributions, whereas those from distinct basins are statistically separable. Drawing on Bayesian optimal classification theory, the authors propose an unsupervised algorithm that bypasses grid discretization and spectral decomposition. The method employs neural networks to approximate the discriminant function and iteratively clusters candidate initial points to reconstruct basin structures. Evaluated on synthetic systems with high-dimensional noise, the approach accurately recovers underlying basins and substantially outperforms existing spectral and clustering techniques, demonstrating the efficacy and superiority of trajectory-distribution-based discrimination for basin identification in high-dimensional stochastic dynamical systems.
📝 Abstract
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.