  While exogenous variables have a major impact on performance improvement in
time series analysis, inter-series correlation and time dependence among them
are rarely considered in the present continuous methods. The dynamical systems
of multivariate time series could be modelled with complex unknown partial
differential equations (PDEs) which play a prominent role in many disciplines
of science and engineering. In this paper, we propose a continuous-time model
for arbitrary-step prediction to learn an unknown PDE system in multivariate
time series whose governing equations are parameterised by self-attention and
gated recurrent neural networks. The proposed model,
\underline{E}xogenous-\underline{g}uided \underline{P}artial
\underline{D}ifferential \underline{E}quation Network (EgPDE-Net), takes
account of the relationships among the exogenous variables and their effects on
the target series. Importantly, the model can be reduced into a regularised
ordinary differential equation (ODE) problem with special designed
regularisation guidance, which makes the PDE problem tractable to obtain
numerical solutions and feasible to predict multiple future values of the
target series at arbitrary time points. Extensive experiments demonstrate that
our proposed model could achieve competitive accuracy over strong baselines: on
average, it outperforms the best baseline by reducing $9.85\%$ on RMSE and
$13.98\%$ on MAE for arbitrary-step prediction.
