Neural reconstruction of Lorenz attractors by an observable

In this paper, Locally Recurrent Neural Networks (LRNNs) are used to learn the chaotic trajectory of the Lorenz system starting from measurements of an observable. An analysis of the Lorenz system based on algebraic observability ensures, in principle, the feasibility of the reconstruction. Thus, algebraic observability of a non-linear system determines the sufficient conditions for the existence of a solution for a state reconstruction problem. Moreover, this procedure can be directly implemented by LRNNs. Results show that LRNNs can be trained to identify and to predict the link among the Lorenz state variables starting from a proper observable, i.e., the LRNNs reconstruct the Lorenz attractor. Furthermore, the LRNNs exhibit generalization properties, i.e., they are characterized by a chaotic Lorenz-like strange attractor under autonomous working conditions.