Differential-Embedding Reconstruction of Dynamical Systems from Scalar Time Series

Abstract

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 Rössler 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 Rössler 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 Rössler systems.