  Upcoming technologies like digital twins, autonomous, and artificial
intelligent systems involving safety-critical applications require models which
are accurate, interpretable, computationally efficient, and generalizable.
Unfortunately, the two most commonly used modeling approaches, physics-based
modeling (PBM) and data-driven modeling (DDM) fail to satisfy all these
requirements. In the current work, we demonstrate how a hybrid approach
combining the best of PBM and DDM can result in models which can outperform
them both. We do so by combining partial differential equations based on first
principles describing partially known physics with a black box DDM, in this
case, a deep neural network model compensating for the unknown physics. First,
we present a mathematical argument for why this approach should work and then
apply the hybrid approach to model two dimensional heat diffusion problem with
an unknown source term. The result demonstrates the method's superior
performance in terms of accuracy, and generalizability. Additionally, it is
shown how the DDM part can be interpreted within the hybrid framework to make
the overall approach reliable.
