  We propose a continuous neural network architecture, termed Explainable
Tensorized Neural Ordinary Differential Equations (ETN-ODE), for multi-step
time series prediction at arbitrary time points. Unlike the existing
approaches, which mainly handle univariate time series for multi-step
prediction or multivariate time series for single-step prediction, ETN-ODE
could model multivariate time series for arbitrary-step prediction. In
addition, it enjoys a tandem attention, w.r.t. temporal attention and variable
attention, being able to provide explainable insights into the data.
Specifically, ETN-ODE combines an explainable Tensorized Gated Recurrent Unit
(Tensorized GRU or TGRU) with Ordinary Differential Equations (ODE). The
derivative of the latent states is parameterized with a neural network. This
continuous-time ODE network enables a multi-step prediction at arbitrary time
points. We quantitatively and qualitatively demonstrate the effectiveness and
the interpretability of ETN-ODE on five different multi-step prediction tasks
and one arbitrary-step prediction task. Extensive experiments show that ETN-ODE
can lead to accurate predictions at arbitrary time points while attaining best
performance against the baseline methods in standard multi-step time series
prediction.
