The phase behavior of polymers in room temperature ionic liquids is a topic of considerable interest. In this work we study the phase diagram of poly(ethylene oxide) in four imidazolium ionic liquids (ILs) using molecular simulation. We develop united atom models for 1-butyl-2,3-dimethylimidazolium ([BMMIM]), 1-ethyl-2,3-dimethylimidazolium GEMMIMD, and 1-ethyl-3-methylimidazolium ([EMIM]) in an analogous fashion to previously developed models for 1-butyl-3-methylimidazolium ([BMIM]) and tetrafluoroborate ([BF4]) using symmetry-adapted perturbation theory. At high temperatures we obtain the coexistence concentrations using an interface method where the polymer and IL are simulated in a large elongated box, and an interface between coexisting phases is formed. At lower temperatures we use a deep neural network (DNN) method. The input descriptors for the DNN are the cohesive energy of mixing, the volume change of mixing, and the coordination numbers between cation and polymer, all of which are obtained from simulations of mixed systems at a series of temperatures. The DNN is trained by using the phase-separated systems at high temperatures and a mixed phase at low temperatures. The method predicts a lower critical solution temperature which decreases as the alkyl chain length on the cation is decreased, consistent with experiment. The simulations show that methylation of the cation has little effect on the phase diagram. This is in contrast to what is seen in experiments but could be because the polymer chains in the simulations are too short. At low temperatures the chains display two conformational motifs, namely a crown ether conformation and a ring conformation, each of which can wrap the chain around a single cation. This provides the entropic penalty for mixing and a reason for demixing as the temperature is raised. Such conformations might not be possible for longer chains. The combination of data-driven techniques and molecular simulation shows promise in the study of the phase behavior and physical properties of complex fluids.