  We consider a power transmission system monitored with Phasor Measurement
Units (PMUs) placed at significant, but not all, nodes of the system. Assuming
that a sufficient number of distinct single-line faults, specifically pre-fault
state and (not cleared) post-fault state, are recorded by the PMUs and are
available for training, we, first, design a comprehensive sequence of Neural
Networks (NNs) locating the faulty line. Performance of different NNs in the
sequence, including Linear Regression, Feed-Forward NN, AlexNet, Graphical
Convolutional NN, Neural Linear ODE and Neural Graph-based ODE, ordered
according to the type and amount of the power flow physics involved, are
compared for different levels of observability. Second, we build a sequence of
advanced Power-System-Dynamics-Informed and Neural-ODE based Machine Learning
schemes trained, given pre-fault state, to predict the post-fault state and
also, in parallel, to estimate system parameters. Finally, third, and
continuing to work with the first (fault localization) setting we design a
(NN-based) algorithm which discovers optimal PMU placement.
