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We study the use of a deep Gaussian process (DGP) prior in a generalnonlinear inverse problem satisfying certain regularity conditions. We provethat when the data arises from a true parameter $\theta^*$ with a compositionalstructure, the posterior induced by the DGP prior concentrates around$\theta^*$ as the number of observations increases. The DGP prior accounts forthe unknown compositional structure through the use of a hierarchical structureprior. As examples, we show that our results apply to Darcy's problem ofrecovering the scalar diffusivity from a steady-state heat equation and theproblem of determining the attenuation potential in a steady-stateSchr\"{o}dinger equation. We further provide a lower bound, proving in Darcy'sproblem that typical Gaussian priors based on Whittle-Mat\'{e}rn processes(which ignore compositional structure) contract at a polynomially slower ratethan the DGP prior for certain diffusivities arising from a generalisedadditive model.

Deep Gaussian Process Priors for Bayesian Inference in Nonlinear Inverse Problems