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We consider a power transmission system monitored with Phasor MeasurementUnits (PMUs) placed at significant, but not all, nodes of the system. Assumingthat a sufficient number of distinct single-line faults, specifically pre-faultstate and (not cleared) post-fault state, are recorded by the PMUs and areavailable for training, we, first, design a comprehensive sequence of NeuralNetworks (NNs) locating the faulty line. Performance of different NNs in thesequence, including Linear Regression, Feed-Forward NN, AlexNet, GraphicalConvolutional NN, Neural Linear ODE and Neural Graph-based ODE, orderedaccording to the type and amount of the power flow physics involved, arecompared for different levels of observability. Second, we build a sequence ofadvanced Power-System-Dynamics-Informed and Neural-ODE based Machine Learningschemes trained, given pre-fault state, to predict the post-fault state andalso, in parallel, to estimate system parameters. Finally, third, andcontinuing to work with the first (fault localization) setting we design a(NN-based) algorithm which discovers optimal PMU placement.

Which Neural Network to Choose for Post-Fault Localization, Dynamic State Estimation and Optimal Measurement Placement in Power Systems?