Reconstruction of nonlinear flows from noisy time series

Nonlinear dynamics is a rapidly developing subject across all disciplines involving spatial or temporal evolution. The reconstruction of equations of motion for a nonlinear system from observed time series has been a hot topic for a long time. Nevertheless, in practice only partial information is available for many systems which are very likely contaminated with noise. Here, based on the invariance of the evolution equation of an autonomous system during time translation, a globally valid local approximation of the trajectory is determined, which could be reliably used for the reconstruction of the vector fields with unknown parameters or functional forms, or even with partial observations. Moreover, the noise interference with nonlinearity is computed to the leading order, which together with the global consideration bestows exceptional robustness and extra accuracy to the technique. The new scheme asks only for the solutions of linear equations and thus is very efficient, which is nicely demonstrated in the Lorenz equation in different conditions, while an FitzHugh–Nagumo (FHN) neural network model is used to show its strength in high dimensions.