An application of atomistic machine learning for variational transition state theory is presented. The rate constants for various radical-radical reactions are computed using variable reaction coordinate transition state theory. In order to simplify the calculation process, artificial neural networks are trained on a structured grid of potential energy data. The resulting surrogate potential energy surface is used in classical phase space representations to describe the interaction between two radical species in the gas phase. When the artificial neural network is trained to potential energy alone, the number of explicit electronic structure energy calculations required to compute a rate constant decreases by at least a factor of 4. When forces are included in the training data, the reduction is more dramatic-at least an order of magnitude.