  Specifying a governing physical model in the presence of missing physics and
recovering its parameters are two intertwined and fundamental problems in
science. Modern machine learning allows one to circumvent these, via emulators
and surrogates, but in doing so disregards prior knowledge and physical laws
that are especially important for small data regimes, interpretability, and
decision making. In this work we fold the mechanistic model into a flexible
data-driven surrogate to arrive at a physically structured decoder network.
This provides accelerated inference for the Bayesian inverse problem, and can
act as a drop-in regulariser that encodes a-priori physical information. We
employ the variational form of the PDE problem and introduce stochastic local
approximations as a form of model based data augmentation. We demonstrate both
the accuracy and increased computational efficiency of the framework on real
world settings and structured spatial processes.
