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

翻译：我们研究了在满足特定正则性条件的一般非线性反问题中采用深度高斯过程（DGP）先验的方法。我们证明，当数据来源于具有组合结构的真实参数θ*时，随着观测数增加，由DGP先验诱导的后验分布会围绕θ*集中。DGP先验通过层级结构先验来刻画未知的组合结构。作为实例，我们展示了该方法可应用于稳态热方程中恢复标量扩散率的达西问题，以及稳态薛定谔方程中确定衰减势的问题。我们进一步给出了下界证明：在达西问题中，对于由广义加性模型生成的特定扩散率，基于Whittle-Matérn过程的典型高斯先验（忽略组合结构）的收缩速率多项式地慢于DGP先验。