  Data-driven surrogate models are widely used for applications such as design
optimization and uncertainty quantification, where repeated evaluations of an
expensive simulator are required. For most partial differential equation (PDE)
simulators, the outputs of interest are often spatial or spatial-temporal
fields, leading to very high-dimensional outputs. Despite the success of
existing data-driven surrogates for high-dimensional outputs, most methods
require a significant number of samples to cover the response surface in order
to achieve a reasonable degree of accuracy. This demand makes the idea of
surrogate models less attractive considering the high computational cost to
generate the data. To address this issue, we exploit the multi-fidelity nature
of a PDE simulator and introduce deep coregionalization, a Bayesian
non-parametric autoregressive framework for efficient emulation of
spatial-temporal fields. To effectively extract the output correlations in the
context of multi-fidelity data, we develop a novel dimension reduction
technique, residual principal component analysis. Our model can simultaneously
capture the rich output correlations and the fidelity correlations and make
high-fidelity predictions with only a few expensive, high-fidelity simulation
samples. We show the advantages of our model in three canonical PDE models and
a fluid dynamics problem. The results show that the proposed method cannot only
approximate a simulator with significantly less cost (at bout 10%-25%) but also
further improve model accuracy.
