Deep Surrogate Accelerated Delayed-Acceptance Hamiltonian Monte Carlo for Bayesian Inference of Spatio-Temporal Heat Fluxes in Rotating Disc Systems

We introduce a deep learning accelerated methodology to solve PDE-based Bayesian inverse problems with guaranteed accuracy. This is motivated by solving the ill-posed problem of inferring a spatio-temporal heat-flux parameter known as the Biot number in a PDE model given temperature data; however, the methodology is generalizable to other settings. To achieve accelerated 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 a 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 nonconvex training. Biot number calculations are involved in turbo-machinery design, which is safety critical and highly regulated, and therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high-dimensional parameter spaces, while 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. A range of numerical results is given using real and simulated data that compare adaptive and general training regimes and various gradient-based Markov chain Monte Carlo sampling methods.