  In this work we propose an extension of physics informed supervised learning
strategies to parametric partial differential equations. Indeed, even if the
latter are indisputably useful in many applications, they can be
computationally expensive most of all in a real-time and many-query setting.
Thus, our main goal is to provide a physics informed learning paradigm to
simulate parametrized phenomena in a small amount of time. The physics
information will be exploited in many ways, in the loss function (standard
physics informed neural networks), as an augmented input (extra feature
employment) and as a guideline to build an effective structure for the neural
network (physics informed architecture). These three aspects, combined
together, will lead to a faster training phase and to a more accurate
parametric prediction. The methodology has been tested for several equations
and also in an optimal control framework.
