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.

翻译：本文提出一种深度学习加速方法，用于在保证精度的前提下求解基于偏微分方程的贝叶斯反问题。该方法源于根据温度数据推断时空热流参数（即毕渥数）这一病态问题，但可推广至其他场景。为加速贝叶斯推断，我们开发了一种新颖的训练方案，利用数据自适应训练神经网络代理模型来模拟参数化正问题。通过同时识别毕渥数的近似后验分布，并根据该分布对物理信息训练损失进行加权，我们的方法无需外部求解即可共同逼近正反问题的解。利用随机切比雪夫级数，我们概述了如何近似高斯过程先验，并借助代理模型应用哈密顿蒙特卡洛方法从后验分布中采样。我们证明了当自适应损失趋近零时，代理后验分布可在Hellinger度量下收敛至真实后验分布。此外，我们阐述了如何将这种代理加速HMC方法与延迟接受方案中的传统PDE求解器相结合，实现后验精度的先验控制。这克服了基于深度学习的代理方法的主要局限性——因其非凸训练特性而无法事先保证精度。毕渥数计算涉及涡轮机械设计这一安全关键且高度受控的领域，因此我们的结果具备此类数学保证至关重要。我们的方法在高维空间中实现快速混合，同时保留传统PDE求解器的收敛保证，且无需对可能被拒绝的建议值进行求解计算。数值结果基于真实与模拟数据给出。