  Substitution of well-grounded theoretical models by data-driven predictions
is not as simple in engineering and sciences as it is in social and economic
fields. Scientific problems suffer most times from paucity of data, while they
may involve a large number of variables and parameters that interact in complex
and non-stationary ways, obeying certain physical laws. Moreover, a
physically-based model is not only useful for making predictions, but to gain
knowledge by the interpretation of its structure, parameters, and mathematical
properties. The solution to these shortcomings seems to be the seamless
blending of the tremendous predictive power of the data-driven approach with
the scientific consistency and interpretability of physically-based models. We
use here the concept of physically-constrained neural networks (PCNN) to
predict the input-output relation in a physical system, while, at the same time
fulfilling the physical constraints. With this goal, the internal hidden state
variables of the system are associated with a set of internal neuron layers,
whose values are constrained by known physical relations, as well as any
additional knowledge on the system. Furthermore, when having enough data, it is
possible to infer knowledge about the internal structure of the system and, if
parameterized, to predict the state parameters for a particular input-output
relation. We show that this approach, besides getting physically-based
predictions, accelerates the training process, reduces the amount of data
required to get similar accuracy, filters partly the intrinsic noise in the
experimental data and provides improved extrapolation capacity.
