A deep surrogate approach to efficient Bayesian inversion in PDE and integral equation models

We investigate a deep learning approach to efficiently perform Bayesian inference in partial differential equation (PDE) and integral equation models over potentially high-dimensional parameter spaces. The contributions of this paper are two-fold; the first is the introduction of a neural network approach to approximating the solutions of Fredholm and Volterra integral equations of the first and second kind. The second is the development of a new, efficient deep learning-based method for Bayesian inversion applied to problems that can be described by PDEs or integral equations. To achieve this we introduce a surrogate model, and demonstrate how this allows efficient sampling from a Bayesian posterior distribution in which the likelihood depends on the solutions of PDEs or integral equations. Our method relies on the direct approximation of parametric solutions by neural networks, without need of traditional numerical solves. This deep learning approach allows the accurate and efficient approximation of parametric solutions in significantly higher dimensions than is possible using classical discretisation schemes. Since the approximated solutions can be cheaply evaluated, the solutions of Bayesian inverse problems over large parameter spaces are efficient using Markov chain Monte Carlo. We demonstrate the performance of our method using two real-world examples; these include Bayesian inference in the PDE and integral equation case for an example from electrochemistry, and Bayesian inference of a function-valued heat-transfer parameter with applications in aviation.