  This paper considers the creation of parametric surrogate models for
applications in science and engineering where the goal is to predict
high-dimensional spatiotemporal output quantities of interest, such as
pressure, temperature and displacement fields. The proposed methodology
develops a low-dimensional parametrization of these quantities of interest
using space-time bases combining with machine learning methods. In particular,
the space-time solutions are sought in a low-dimensional space-time linear
trial subspace that can be obtained by computing tensor decompositions of usual
state-snapshots data. The mapping between the input parameters and the basis
expansion coefficients (or generalized coordinates) is approximated using four
different machine learning techniques: multivariate polynomial regression,
k-nearest-neighbors, random forest and neural network. The relative costs and
effectiveness of the four machine learning techniques are explored through
three engineering problems: steady heat conduction, unsteady heat conduction
and unsteady advective-diffusive-reactive system. Numerical results demonstrate
that the proposed method performs well in terms of both accuracy and
computational cost, and highlight the important point that the amount of model
training data available in an engineering setting is often much less than it is
in other machine learning applications, making it essential to incorporate
knowledge from physical models. In addition, simpler machine learning
techniques are seen to perform better than more elaborate ones.
