Numerical methods are very important in geotechnical and geological engineering. This study presents a reduced-order numerical model to approximate the displacement and stress field in geotechnical and geological engineering contexts by combining numerical methods, proper orthogonal decomposition (POD), and multi-output support vector machine (MSVM). Snapshots were generated using Latin hypercube sampling. POD was used to compute POD-based vectors and their coefficients. Training samples were constructed from the numerical model and POD coefficients input. The MSVM algorithm was adopted to present the relationship based on these training samples. A reduced-order model was developed by predicting the POD coefficients using MSVM, and the displacement and stress field was then predicted based on the POD vectors and predicted POD coefficients. The proposed method was verified and demonstrated for a circular tunnel. The results show the displacement and stress fields are in excellent agreement with the analytical solution and with the numerical solution, and the predicted deformations are consistent with rock mechanics theory. The proposed method predicts the deformation and mechanical behavior of geomaterials well and may be used to replace numerical models for back analysis, for optimal design, and for uncertainty analysis in geotechnical and geological engineering, all of which involve repeated computation.