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.
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 present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.
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
Given a scalar observable of an ergodic dynamical system with a low-dimensional attractor, two families of methods reconstruct and predict the underlying state: recurrence-based methods (the method of analogues and its descendants), which wait for the trajectory to return to an $\varepsilon$-neighborhood of a previously observed state, and observer-based methods, which fit a converging state estimator on the delay reconstruction. We formalize and empirically verify an exponential separation between the two: the expected cost of recurrence scales as $\varepsilon^{-d}$, where $d$ is the pointwise dimension of the invariant measure (a consequence of the Kac lemma and quantitative Poincare recurrence), whereas a detectable linear observer converges in $\Theta(\log(1/\varepsilon)/(1-\rho(A_{cl})^2))$ steps, where $\rho(A_{cl})$ is the closed-loop spectral radius of the Riccati fixed point. Both laws are verified numerically (return-time exponent $-1.8$ on the Lorenz attractor against the theoretical $-2.05$; observer cost linear in $\log(1/\varepsilon)$ with $R^2=1.000$ and in $(1-\rho^2)^{-1}$ with $R^2=0.985$), yielding a measured cost gap of $\sim 10^{9}$ at $\varepsilon=10^{-6}$ for $d\approx 2$. We complement the theorem with an admission protocol (the Kac-Riccati gate) deciding whether a signal lies inside the theorem's class, via surrogate-data prediction gating; it also explains the folklore of"universal"fractal dimensions as a dataset-size artifact bounded by $2\log_{10}N$. On real data the gate admits the Santa Fe laser benchmark ($\hat D_2=2.0$) and refuses the monthly sunspot series, reproducing the settled resolution of historical low-dimensionality claims. All results reproduce from a single verification script (17/17 checks).
The data-driven discovery of governing equations for dynamical systems has emerged as a transformative paradigm, enabling the extraction of interpretable and generalizable models from observational data. While modern techniques have advanced this field, traditional subset regression remains a foundational yet underutilized tool due to its reliance on uncorrelated residuals, a requirement often violated by time-series data. In this work, we revisit subset regression to identify dynamical systems governed by ordinary differential equations (ODEs), partial differential equations (PDEs), and differential algebraic equations (DAEs). We propose subset regression with known number of active features (sub-KNAFE), a user-determined sparsity mechanism that flexibly adapts to various complex nonlinear systems, while retaining the computational efficiency and inherent interpretability of traditional subset regression. We integrate sub-KNAFE with the SINDy framework, overcoming the limitation of subset regression in dynamical system identification. Numerical tests across a range of signal-to-noise ratios and dataset sizes demonstrate sub-KNAFE's superior noise robustness and data efficiency. Practical utility for sub-KNAFE is validated on two real-world datasets: the classic Lynx-Hare ecological population data and the ISO New England power system dataset, demonstrating its strong potential for practical deployment in scientific discovery and engineering applications.
Weizhen Li, Qiang Fu, Yifan Hong et al.· Scientific Reports· 0 citations
This work first learns an implicit spectral predictor using Observation Spectral Filtering using Observation Spectral Filtering, a convex method that competes with the best linear observer for the system, and applies spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system.
Liane Galanti, Devan Shah, Shlomo Fortgang et al.· 0 citations