  Physical simulations based on partial differential equations typically
generate spatial fields results, which are utilized to calculate specific
properties of a system for engineering design and optimization. Due to the
intensive computational burden of the simulations, a surrogate model mapping
the low-dimensional inputs to the spatial fields are commonly built based on a
relatively small dataset. To resolve the challenge of predicting the whole
spatial field, the popular linear model of coregionalization (LMC) can
disentangle complicated correlations within the high-dimensional spatial field
outputs and deliver accurate predictions. However, LMC fails if the spatial
field cannot be well approximated by a linear combination of base functions
with latent processes. In this paper, we present the Extended Linear Model of
Coregionalization (E-LMC) by introducing an invertible neural network to
linearize the highly complex and nonlinear spatial fields so that the LMC can
easily generalize to nonlinear problems while preserving the traceability and
scalability. Several real-world applications demonstrate that E-LMC can exploit
spatial correlations effectively, showing a maximum improvement of about 40%
over the original LMC and outperforming the other state-of-the-art spatial
field models.
