  We introduce a deep learning accelerated methodology to solve PDE-based
Bayesian inverse problems with guaranteed accuracy. This is motivated by the
ill-posed problem of inferring a spatio-temporal heat-flux parameter known as
the Biot number given temperature data, however the methodology is
generalisable to other settings. To accelerate Bayesian inference, we develop a
novel training scheme that uses data to adaptively train a neural-network
surrogate simulating the parametric forward model. By simultaneously
identifying an approximate posterior distribution over the Biot number, and
weighting a physics-informed training loss according to this, our approach
approximates forward and inverse solution together without any need for
external solves. Using a random Chebyshev series, we outline how to approximate
a Gaussian process prior, and using the surrogate we apply Hamiltonian Monte
Carlo (HMC) to sample from the posterior distribution. We derive convergence of
the surrogate posterior to the true posterior distribution in the Hellinger
metric as our adaptive loss approaches zero. Additionally, we describe how this
surrogate-accelerated HMC approach can be combined with traditional PDE solvers
in a delayed-acceptance scheme to a-priori control the posterior accuracy. This
overcomes a major limitation of deep learning-based surrogate approaches, which
do not achieve guaranteed accuracy a-priori due to their non-convex training.
Biot number calculations are involved in turbo-machinery design, which is
safety critical and highly regulated, therefore it is important that our
results have such mathematical guarantees. Our approach achieves fast mixing in
high dimensions whilst retaining the convergence guarantees of a traditional
PDE solver, and without the burden of evaluating this solver for proposals that
are likely to be rejected. Numerical results are given using real and simulated
data.
