  Bayesian Inference with Markov Chain Monte Carlo requires efficient
computation of the likelihood function. In some scientific applications, the
likelihood must be computed by numerically solving a partial differential
equation, 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. We show that this has two
additional benefits: reducing noise in the likelihood evaluations and providing
fast gradient calculations. In experiments, the approach is applied to a model
of heliospheric transport of galactic cosmic rays, where it enables efficient
sampling from the posterior of latent parameters in the Parker equation.
