Yi Zhang, Ph.D., presented "Bayesian inference with partial differential equations using Stan" at StanCon in Helsinki, Finland on August 30, 2018. You can watch video of his presentation here. You can see his poster and slide presentation on our website www.metrumrg.com.
Abstract: We propose an interface for Stan language to incorporate user-defined PDE solution routine to perform Bayesian inference that involves PDEs. We demonstrate various approaches of obtaining sensitivities with respect to PDE parameters using numerical examples.