  Parameter Estimation (PE) and State Estimation (SE) are the most wide-spread
tasks in the system engineering. They need to be done automatically, fast and
frequently, as measurements arrive. Deep Learning (DL) holds the promise of
tackling the challenge, however in so far, as PE and SE in power systems is
concerned, (a) DL did not win trust of the system operators because of the lack
of the physics of electricity based, interpretations and (b) DL remained
illusive in the operational regimes were data is scarce. To address this, we
present a hybrid scheme which embeds physics modeling of power systems into
Graphical Neural Networks (GNN), therefore empowering system operators with a
reliable and explainable real-time predictions which can then be used to
control the critical infrastructure. To enable progress towards trustworthy DL
for PE and SE, we build a physics-informed method, named Power-GNN, which
reconstructs physical, thus interpretable, parameters within Effective Power
Flow (EPF) models, such as admittances of effective power lines, and NN
parameters, representing implicitly unobserved elements of the system. In our
experiments, we test the Power-GNN on different realistic power networks,
including these with thousands of loads and hundreds of generators. We show
that the Power-GNN outperforms vanilla NN scheme unaware of the EPF physics.
