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We introduce a deep learning accelerated methodology to solve PDE-basedBayesian inverse problems with guaranteed accuracy. This is motivated by theill-posed problem of inferring a spatio-temporal heat-flux parameter known asthe Biot number given temperature data, however the methodology isgeneralisable to other settings. To accelerate Bayesian inference, we develop anovel training scheme that uses data to adaptively train a neural-networksurrogate simulating the parametric forward model. By simultaneouslyidentifying an approximate posterior distribution over the Biot number, andweighting a physics-informed training loss according to this, our approachapproximates forward and inverse solution together without any need forexternal solves. Using a random Chebyshev series, we outline how to approximatea Gaussian process prior, and using the surrogate we apply Hamiltonian MonteCarlo (HMC) to sample from the posterior distribution. We derive convergence ofthe surrogate posterior to the true posterior distribution in the Hellingermetric as our adaptive loss approaches zero. Additionally, we describe how thissurrogate-accelerated HMC approach can be combined with traditional PDE solversin a delayed-acceptance scheme to a-priori control the posterior accuracy. Thisovercomes a major limitation of deep learning-based surrogate approaches, whichdo not achieve guaranteed accuracy a-priori due to their non-convex training.Biot number calculations are involved in turbo-machinery design, which issafety critical and highly regulated, therefore it is important that ourresults have such mathematical guarantees. Our approach achieves fast mixing inhigh dimensions whilst retaining the convergence guarantees of a traditionalPDE solver, and without the burden of evaluating this solver for proposals thatare likely to be rejected. Numerical results are given using real and simulateddata.

Deep surrogate accelerated delayed-acceptance HMC for Bayesian inference of spatio-temporal heat fluxes in rotating disc systems