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.
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.
Multi-step training of sparse, interpretable models of dynamical systems directly from time-series data yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents.
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...
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 previousl...
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 underutil...
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 syst...
Liane Galanti, Devan Shah, Shlomo Fortgang et al.· 0 citations
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