The accurate identification of internal and external pressures in thick-walled hyperelastic vessels is a challenging inverse problem with significant implications for structural health monitoring, biomedical devices, and soft robotics. Conventional analytical and numerical approaches address the forward problem effectively but offer limited means for recovering unknown load conditions from observable deformations. In this study, we introduce a Graph-FEM/ML framework that couples high-fidelity finite element simulations with machine learning models to infer normalized internal and external pressures from measurable boundary deformations. A dataset of 1386 valid samples was generated through Latin Hypercube Sampling of geometric and loading parameters and simulated using finite element analysis with a Neo-Hookean constitutive model. Two complementary neural architectures were explored: graph neural networks (GNNs), which operate directly on resampled and feature-enriched boundary data, and convolutional neural networks (CNNs), which process image-based representations of undeformed and deformed cross-sections. The GNN models consistently achieved low root-mean-square errors (≈0.021) and stable correlations across training, validation, and test sets, particularly when augmented with displacement and directional features. In contrast, CNN models exhibited limited predictive accuracy: quarter-section inputs regressed toward mean values, while full-ring and filled-section inputs improved after Bayesian optimization but remained inferior to GNNs, with higher RMSEs (0.023–0.030) and modest correlations (R2). To the best of our knowledge, this is the first work to combine boundary deformation observations with graph-based learning for inverse load identification in hyperelastic vessels. The results highlight the advantages of boundary-informed GNNs over CNNs and establish a reproducible dataset and methodology for future investigations. This framework represents an initial step toward a new direction in mechanics-informed machine learning, with the expectation that future research will refine and extend the approach to improve accuracy, robustness, and applicability in broader engineering and biomedical contexts.