Fig. 1. Schematic illustration of the auto-reservoir neural network.

a Given a short-term time series of a high-dimensional system, it is a challenging task to predict future states of any target variable. For a target variable y to be predicted, a delay-embedding strategy is applied, forming a delayed-coordinate vector Y^t corresponding to the observed vector X^t via a function Φ. Such a relation constitutes the spatiotemporal information (STI) transformation with both primary and conjugate forms (STI equations). b The linearized STI equations also have primary and conjugate forms. Data can be represented in a matrix form where the future/unknown information {$y^{m+1},y^{m+2},\dots ,y^{m+L-1}$} is located in the lower-right triangle of matrix Y and the known information {y¹, y², …, y^m} in the upper-left part of Y. c Auto-reservoir neural network (ARNN) is a model-free method to make the multistep-ahead prediction for a target y. In the ARNN framework, the reservoir component contains a random/fixed multilayer neural network F, for which there are time-dependent inputs X^t. A target vector Y^t formed by the delay embedding for the prediction is processed through neural network F with two weight matrices A and B. Such an architecture of ARNN is designed to simultaneously solve both primary and conjugate forms of ARNN-based STI equations to enhance the robustness, thus predicting the future information of the target variable y even with a short-term time series. d According to the information flow, ARNN has an autoencoder-like framework, that is, $F\left({\mathbf{X}}^t\right)\to {\mathbf{Y}}^t\to F\left({\mathbf{X}}^t\right)$, different from but similar to the autoencoder structure ${\mathbf{X}}^t\to {\mathbf{Y}}^t\to {\mathbf{X}}^t$.