An algorithmic framework, TAkens Reconstruction (TAR), to analyze and reconstruct arbitrary dynamical systems from low-dimensional time series using an integration of Takens'Delay Embedding Theorem, manifold learning techniques, and universal function approximators is developed.
Abstract
Embedding theorems can be used to provide theoretical guarantees about the relation between low-dimensional observations of a system and its full-dimensional state and dynamics. Such theorems do not, however, provide guidance on observable choice, embedding construction, or methodologies to learn the mapping between the embedding and full-dimensional state. In this work, we develop an algorithmic framework, TAkens Reconstruction (TAR), to analyze and reconstruct arbitrary dynamical systems from low-dimensional time series using an integration of Takens'Delay Embedding Theorem, manifold learning techniques, and universal function approximators. We validate TAR in applications to a variety of simulated and observed dynamical systems and use it to investigate how delay vector structure impacts reconstruction accuracy. In an ecological system, we show that simple predator-prey dynamics can be reconstructed with observations taken over a wide variety of embedding time scales. In molecular dynamics simulations of the protein Villin, we demonstrate how including multiple time delays of the same observable series can be used to improve reconstruction of systems with multiple characteristic time scales. In the trade record of Vanguard S&P 500, we show how the approach exposes underlying dynamical phenomenologies in the data and accurate return predictions over short time horizons without access to full-dimensional market observations. We develop and release an open-source software package to enable the application of TAR to arbitrary dynamical systems.
We study the reconstruction of an unknown dynamical system from a single noisy scalar time series. The goal is to recover the underlying dynamics for forecasting. We introduce a method that uses differential embedding coordinates to identify a rational closure of the embedding dynamics directly from data. The closure is identified through a weak-form regression pipeline, which avoids unstable pointwise differentiation of noisy data. When applied to noise-free Lorenz and R\"ossler systems, the method recovers closures that support long forecasts across a broad ensemble of realizations ($18.1$ and $7.1$ Lyapunov times respectively). Under $15$--$30\%$ additive Gaussian noise, performance becomes system-dependent. For the Lorenz system, forecast horizons remain short even in the best cases, whereas the R\"ossler system generally performs better in absolute terms, though not once normalized by the Lyapunov time. Our proposed method recovers directly interpretable closure coefficients which we compared against the known analytic closures of the Lorenz and R\"ossler systems.
Takens'time-delay embedding theorem provides conditions under which delay-coordinate maps, formed using uniformly-sampled time series of trajectories evolving on attractors of dynamical systems, can faithfully represent the dynamics of the original system. Nonlinear systems can be highly sensitive, and Takens'theorem does not provide guarantees about the stability of time-delay embeddings. In the linear setting, statements about the stability of time-delay embeddings are more tractable and have been proven for delay-coordinate maps with evenly-spaced delays. In many experimental applications, however, time series data may be non-uniformly-sampled, especially in systems with multiple timescales or when using event-based rather than time-based sampling techniques. In this paper, we extend the theorems for the stable linear Takens'embeddings to the setting where the delay-coordinate maps involve unevenly-spaced delays. We pose a conjecture about the rank of generalized Vandermonde matrices that capture the temporal structure of time-delay embeddings. We prove that, provided the conjecture holds, existing theorems about stable linear Takens'embeddings readily extend to unevenly-sampled settings, and the quality of the embedding converges to the same asymptotic bounds when using a large number of delays in the delay-coordinate map, a result which is supported by numerical simulations.
By focusing on algorithmic stability as a means of establishing out-of-sample bounds, we provide a system-theoretic interpretation of generalization in learning-enabled dynamical systems arising in data-driven optimization and feedback control approximation. Given two neighboring datasets, we specifically model sample replacement as an exogenous disturbance acting on a sensitivity system, while the incremental behavior of the data-dependent operator is encoded through an integral quadratic constraint. By relying on dissipativity arguments, we establish a matrix inequality-based certificate and a uniform stability bound that separates the one-sample sensitivity of the learned operator, and an algorithm-dependent dynamical gain. The latter can then be optimized, offering a tractable tool for certifying and comparing generalization capabilities of learning dynamics. We show that our results recover classical ones for gradient descent, apply naturally to momentum-based methods such as heavy-ball and Nesterov acceleration, and extend to data-driven control.
Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.
Markus Heinonen, Yair Shenfeld, R. Baptista et al.· 0 citations
In the framework of network dynamics, learning models, and neural tangent kernels (NTK), we show that the corresponding linearized dynamics leads naturally to a semigroup formulation. More precisely, in our analysis of input/output models, the time-dynamics is presented via special semigroups of linear operators on Hilbert spaces, together with an associated class of semigroup perturbations. In this context, we then present new and explicit a priori perturbation-bound results: for the fixed-kernel linearization constructions arising in the NTK setting, we prove norm-bounds on the corresponding semigroup perturbations, in the form of explicit finite-time perturbation estimates. We further present refinements on prescribed task spaces, Ces\`aro-averaged (ergodic) comparisons estimates, and versions in which the lower spectral edge assumption is replaced by a spectral-distribution condition. We also extend the comparison to nonautonomous NTK evolutions through piecewise-frozen approximations, record a corresponding discrete Euler specialization, and offer worked examples in order to illustrate our perturbation-bound estimates.
Halyun Jeong, Palle E. T. Jorgensen, Hyun-Kyoung Kwon et al.· 0 citations
Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
Rishav Jha, Kameshwar Sahani, S. K. Sahani et al.· African Multidisciplinary Jo...· 0 citations