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The paper proposes an artificial neural network (ANN)-based strategy for identification of reduced-order dynamic equivalents of power systems. This large-signal model is formulated in continuous-time and is therefore compatible with standard models of power system components. In a departure from previous works on the subject, we do not postulate a particular model structure for the equivalent, hence the label nonparametric. The approach uses only measurements at points where internal (retained) and external (reduced) systems are interfaced, and requires no knowledge of parameters and topology of the external subsystem. The procedure consists of two conceptual steps: (1) the first ("bottleneck") ANN is used to extract "states" of the reduced-order equivalent; and (2) the second (recurrent) ANN is embedded in an ordinary differential equations (ODEs) solver, and trained to approximate the "right-hand side," using the states extracted at the first step. We also describe an extension in which a third ANN is used to synthesize missing interface measurements from a historical database of system responses to various disturbances. We illustrate the capabilities of the approach on a multimachine benchmark example derived from the WSCC system.