  The concept of integrating physics-based and data-driven approaches has
become popular for modeling sustainable energy systems. However, the existing
literature mainly focuses on the data-driven surrogates generated to replace
physics-based models. These models often trade accuracy for speed but lack the
generalisability, adaptability, and interpretability inherent in physics-based
models, which are often indispensable in the modeling of real-world dynamic
systems for optimization and control purposes. In this work, we propose a novel
architecture for generating model-integrated neural networks (MINN) to allow
integration on the level of learning physics-based dynamics of the system. The
obtained hybrid model solves an unsettled research problem in control-oriented
modeling, i.e., how to obtain an optimally simplified model that is physically
insightful, numerically accurate, and computationally tractable simultaneously.
We apply the proposed neural network architecture to model the electrochemical
dynamics of lithium-ion batteries and show that MINN is extremely
data-efficient to train while being sufficiently generalizable to previously
unseen input data, owing to its underlying physical invariants. The MINN
battery model has an accuracy comparable to the first principle-based model in
predicting both the system outputs and any locally distributed electrochemical
behaviors but achieves two orders of magnitude reduction in the solution time.
