The drag coefficient $C_d$ plays a central role in the dynamics modeling of particle-fluid interactions in multiphase flows, including particle transport, sedimentation, and multiphase reactors. Although reliable correlations for the drag coefficient have been well established for spherical and regularly shaped particles, predicting $C_d$ for irregularly shaped particles remains a major challenge in particle-fluid system modeling. This difficulty arises because hydrodynamic forces are highly sensitive to fine geometric features, which are not adequately captured by conventional shape descriptors. In this work, we introduce a data-driven framework that leverages a Variational Autoencoder (VAE) to derive a low-dimensional shape representation for predicting the drag coefficient of irregular particles. Unlike conventional manual geometric descriptors, our approach encodes 3D particle geometries into a compact latent space that emphasizes features governing hydrodynamic response. The latent space is incorporated into a supervised regression model to map raw geometry to $C_d$. By replacing manual feature engineering with this learned surrogate representation, this framework achieves a continuous and differentiable mapping from shape to drag behavior across the low-Reynolds-number regime. The feasibility and accuracy of the proposed approach are first evaluated using a dataset of regular ellipsoidal particles, where systematic shape variations allow controlled validation. The method is then extended to two datasets of shape randomly perturbed ellipsoids with increasing degrees of geometric complexity, which provides insight into the generalization performance of the proposed framework. The results demonstrate that low-dimensional shape representations can effectively capture the influence of particle geometry on drag, offering a promising framework for $C_d$ modeling of hydrodynamic forces in particle-laden flows involving irregular particles.