analyze time-derivative spaces

Conventional high-dimensional analysis methods struggle to simultaneously capture the coupling between structure and dynamics in complex many-body systems. This work proposes the Time Derivative (TiDe) space framework, which constructs higher-order time derivatives directly from temporal observational data as intrinsic dimensions, enabling joint unsupervised learning of structure and dynamics without dimensionality reduction. By naturally embedding physical information into high-dimensional representations, TiDe achieves both interpretability and computational robustness. Experiments on molecular dynamics simulations and experimental trajectory data demonstrate that TiDe efficiently uncovers key dynamical patterns in complex systems, significantly outperforming existing approaches.

Prior work lacks a rigorous theoretical foundation linking visual time-series clustering (e.g., TimeCluster) with classical linear subspace identification methods such as Hankel matrix singular value decomposition (SVD). Method: We establish a formal equivalence between the TimeCluster–PCA pipeline and Hankel SVD by proving that the principal component subspace obtained via sliding-window PCA is mathematically identical to the dominant left singular subspace of the corresponding Hankel matrix. Contribution/Results: This is the first strict proof of subspace equivalence, unifying clustering-based and system-identification paradigms; it enables direct interpretation of clustering coordinates as state-space basis vectors. The unified framework integrates sliding-window embedding, PCA, SVD, and Hankel theory, supporting state-space forecasting, online analysis, and noise-robust visualization. Empirical validation on synthetic and real-world dynamical signals confirms embedding consistency, providing a theoretical foundation for interpretable temporal structure discovery.

Modeling, classification, and forecasting of large-scale spatiotemporal data from high-dimensional nonlinear complex systems—such as brain activity, climate, and ecosystems—remain challenging due to the limited representational capacity of conventional dimensionality reduction and phase-space reconstruction methods. Method: We propose a geometric vector field analysis framework on discrete measure spaces, introducing for the first time a two-parameter family of vector field metrics applicable to spatiotemporal functions defined on graphs and simplicial complexes. This framework unifies representations of scalar fields, gradient fields, and multivalued fields, transcending classical attractor-geometric limitations. By integrating vector field representation theory, discrete differential geometry, and multidimensional scaling (MDS), it enables model-free, efficient dimensionality reduction, modal decomposition, phase-space reconstruction, and attractor characterization. Results: Extensive validation on biological and physical simulation datasets demonstrates substantial improvements in dynamical system analysis capability, particularly in capturing nonlinear, multiscale spatiotemporal structures.

This study addresses the challenge of monitoring dynamic processes in high-dimensional time series by proposing a novel approach that integrates topological data analysis (TDA) with neural ordinary differential equations (Neural ODEs). The method represents multivariate time series as manifolds, employs topological descriptors to capture their structural properties, and leverages Neural ODEs to model the continuous evolution of the system’s topology, enabling trajectory-based real-time event detection. To the best of our knowledge, this work is the first to combine TDA with Neural ODEs for process monitoring, overcoming limitations of conventional reconstruction- or Koopman-based methods in capturing topological changes in high-dimensional dynamics. Experimental results on industrial datasets demonstrate that the proposed method significantly outperforms baseline approaches—including PCA, autoencoders, and Koopman autoencoders—across multiple event detection tasks.

This paper addresses the joint clustering and linear dynamical system (LDS) modeling of multiple trajectories: given a trajectory set and a prescribed number of clusters, it simultaneously partitions trajectories and learns an LDS model per cluster to minimize the maximum prediction error across all models. Methodologically, we propose a unified optimization framework that does not require pre-specifying the latent state dimension, integrating globally convergent optimization with an EM-inspired heuristic to jointly solve trajectory assignment and LDS parameter estimation. Theoretically, we derive a provably tight upper bound on regularization selection for system identification. Experiments demonstrate robust convergence and high modeling accuracy, significantly outperforming existing sequential baselines on both synthetic and real-world datasets, thereby validating the method’s effectiveness and practicality.

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.

This work addresses the limitation of conventional dynamical system operator estimation methods, which treat each system in isolation and fail to exploit shared dynamical structures. The authors propose DOODL, a novel framework that integrates dictionary learning with optimal transport for the first time, under the assumption that related systems approximately lie on a low-dimensional manifold in the spectral operator space. By learning a shared basis of spectral dynamical atoms, DOODL geometrically models this manifold to yield compact, interpretable system embeddings and effectively regularizes operator estimation from short or partial observations. Evaluated on Langevin dynamics and turbulent plasma simulations, DOODL reduces estimation errors by one to two orders of magnitude compared to traditional approaches under low-data regimes, substantially improving the recovery accuracy of dominant spectral dynamical structures.

This work addresses the limitations of traditional system identification methods in heterogeneous environments, which rely on iterative clustering and are thus sensitive to model initialization and learning uncertainty. The authors propose a training-free, one-shot clustering approach that quantifies dynamic similarity among systems by measuring the alignment of principal subspaces derived from state covariance matrices estimated using local observational data. By leveraging the intrinsic subspace structure of these covariance matrices, the method circumvents iterative optimization and provides theoretical guarantees on clustering success under finite-sample regimes. Experimental results demonstrate that the proposed approach effectively identifies systems sharing common dynamics, significantly reduces estimation error for personalized models, and outperforms both training-based clustering and non-clustering baselines.