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High-fidelity computer simulations of childbirth remain prohibitively expensive and time consuming, making them impractical for guiding decision-making during obstetric emergencies. Cheap computer simulations that preserve the accuracy of high-fidelity models can be developed using surrogate modeling. Two common approaches to surrogate modeling are physics-based reduced order modeling (ROM) and machine learning (ML), with the latter gaining popularity as the scientific computing community seeks to leverage advances from other, mostly non-physics-based, computational strategies. Although ROM and ML have been compared for various problems, to our knowledge, such a comparison for simulations of vaginal deformations is currently missing. This study provides a baseline numerical comparison between methods from these two fundamentally different approaches. Since there are many methods falling into each modeling approach, to provide a fair and natural comparison, we select a basic model from each category, with each allowing (i) a straightforward implementation in commercial software packages, and (ii) use by practitioners with limited experience in the field. As a benchmark for the numerical comparison of the ROM and ML approaches, we use the finite element (FE) modeling of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. From the ROM strategies, we consider a simplified Galerkin ROM (G-ROM) that is based on the linearization of the underlying nonlinear equations. From the ML strategies, we select a feed-forward neural network to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the proper orthogonal decomposition (POD) coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical investigation of G-ROM, ML, and POD-ML takes place in the reconstructive regime. The numerical results show that the G-ROM outperforms the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The G-ROM achieves superior error performance to the best ML model with 11 POD basis functions. With higher-dimensional POD bases, the G-ROM achieves a relative error 3 orders of magnitude lower than that of the best ML model with an online CPU time still on the same order of magnitude as the best ML model. The POD-ML model improves on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model does not improve on the error performance of the ML and is still outperformed by the G-ROM for POD bases of size greater than 11. This baseline numerical investigation serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, and material properties of the human vagina, as well as their changes during labor. [ABSTRACT FROM AUTHOR]