We present a machine-learning-based particle-identification study for the proximity-focusing Ring Imaging Cherenkov (pfRICH) detector of the ePIC experiment at the Electron-Ion Collider. Operating in the backward region ($-3.5 \lesssim \eta \lesssim -1.5$), the pfRICH is designed to achieve at least $3\sigma$ separation among pions, kaons, and protons up to $7,\mathrm{GeV}/c$ for Semi-Inclusive Deep Inelastic Scattering measurements. Using a standalone Geant4 simulation of the pfRICH, we develop a hybrid machine-learning approach that combines convolutional neural-network-based feature extraction with gradient-boosted decision-tree classifiers. This method significantly enhances Cherenkov-ring pattern recognition and improves particle-separation performance, demonstrating the effectiveness of hybrid machine-learning techniques for next-generation Cherenkov detectors at the EIC.