Bayesian Inference with Markov Chain Monte Carlo requires the ability to efficiently compute the likelihood function. In some scientific applications, the likelihood can only be computed by a numerical PDE solver, which can be prohibitively expensive. We demonstrate that some such problems can be made tractable by amortizing the computation with a surrogate likelihood function implemented by a neural network. This can have the added benefits of reducing noise in the likelihood evaluations and providing fast gradient calculations. We demonstrate these advantages in a model of heliospheric transport of galactic cosmic rays, where our approach enables us to estimate the posterior of five latent parameters of the Parker equation.