  Making accurate forecasts for a complex system is a challenge in various
practical applications. The major difficulty in solving such a problem concerns
nonlinear spatiotemporal dynamics with time-varying characteristics. Takens'
delay embedding theory provides a way to transform high-dimensional spatial
information into temporal information. In this work, by combining delay
embedding theory and deep learning techniques, we propose a novel framework,
Delay-Embedding-based Forecast Machine (DEFM), to predict the future values of
a target variable in a self-supervised and multistep-ahead manner based on
high-dimensional observations. With a three-module spatiotemporal architecture,
the DEFM leverages deep neural networks to effectively extract both the
spatially and temporally associated information from the observed time series
even with time-varying parameters or additive noise. The DEFM can accurately
predict future information by transforming spatiotemporal information to the
delay embeddings of a target variable. The efficacy and precision of the DEFM
are substantiated through applications in three spatiotemporally chaotic
systems: a 90-dimensional (90D) coupled Lorenz system, the Lorenz 96 system,
and the Kuramoto-Sivashinsky (KS) equation with inhomogeneity. Additionally,
the performance of the DEFM is evaluated on six real-world datasets spanning
various fields. Comparative experiments with five prediction methods illustrate
the superiority and robustness of the DEFM and show the great potential of the
DEFM in temporal information mining and forecasting
