  In this work, we introduce, justify and demonstrate the Corrective Source
Term Approach (CoSTA) -- a novel approach to Hybrid Analysis and Modeling
(HAM). The objective of HAM is to combine physics-based modeling (PBM) and
data-driven modeling (DDM) to create generalizable, trustworthy, accurate,
computationally efficient and self-evolving models. CoSTA achieves this
objective by augmenting the governing equation of a PBM model with a corrective
source term generated using a deep neural network. In a series of numerical
experiments on one-dimensional heat diffusion, CoSTA is found to outperform
comparable DDM and PBM models in terms of accuracy -- often reducing predictive
errors by several orders of magnitude -- while also generalizing better than
pure DDM. Due to its flexible but solid theoretical foundation, CoSTA provides
a modular framework for leveraging novel developments within both PBM and DDM.
Its theoretical foundation also ensures that CoSTA can be used to model any
system governed by (deterministic) partial differential equations. Moreover,
CoSTA facilitates interpretation of the DNN-generated source term within the
context of PBM, which results in improved explainability of the DNN. These
factors make CoSTA a potential door-opener for data-driven techniques to enter
high-stakes applications previously reserved for pure PBM.
