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We report on the application of machine learning (ML) methods for predicting the longitudinal phase space (LPS) distribution of particle accelerators. Our approach consists of training a ML-based virtual diagnostic to predict the LPS using only nondestructive linac and e-beam measurements as inputs. We validate this approach with a simulation study for the FACET-II linac and with an experimental demonstration conducted at LCLS. At LCLS, the e-beam LPS images are obtained with a transverse deflecting cavity and used as training data for our ML model. In both the FACET-II and LCLS cases we find good agreement between the predicted and simulated/measured LPS profiles, an important step towards showing the feasibility of implementing such a virtual diagnostic on particle accelerators in the future.