Exoplanet atmospheric retrieval pairs a radiative transfer (RT) model with a Bayesian framework to infer the atmospheric properties of an exoplanet. This technique’s runtime is typically dominated by the RT calculations, which can take seconds per model evaluation depending on model complexity (e.g., spectral resolution, opacity sources, clouds). When executing many retrievals (e.g., to plan for observations or future instruments/telescopes), the computational cost can become significant. Neural networks (NNs) can offer a fast, accurate approximation to a complex process, such as RT, after training on a sufficient data set. By using an NN surrogate model in place of an RT code, retrieval compute costs can be reduced by orders of magnitude. We demonstrate our technique on observations of HD 189733 b. The Bhattacharyya coefficients between the 1D marginalized posteriors of the NN-accelerated retrieval and the Bayesian Atmospheric Radiative Transfer (BART) code are >0.9647, with a mean of 0.9905. We find a reduction in forward model compute time by a factor of ~20 per parallel chain when using an Intel i7-4770 central processing unit (CPU). By utilizing an Nvidia Titan Xp graphics processing unit, forward model compute time is reduced by a factor of ~275 per chain compared to that CPU. Our open-source implementation can be readily applied to other problems beyond exoplanet retrieval. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research. This research was supported by the NASA Fellowship Activity under NASA Grant 80NSSC20K0682 and NASA Exoplanets Research Program grant NNX17AB62G.