This paper proposes a machine learning method to predict the solutions of related nonlinear optimal control problems given some parametric input, such as the initial state. The map between problem parameters to optimal solutions is called the problem-optimum map, and is often discontinuous due to nonconvexity, discrete homotopy classes, and control switching. This causes difficulties for traditional function approximators such as neural networks, which assume continuity of the underlying function. This paper proposes a mixture of experts (MoE) model composed of a classifier and several regressors, where each regressor is tuned to a particular continuous region. A novel training approach is proposed that trains classifier and regressors independently. MoE greatly outperforms standard neural networks, and achieves highly reliable trajectory prediction (over 99.5% accuracy) in several dynamic vehicle control problems.