We consider non-linear Bayesian inverse problems of determining the parameter $f$. For the posterior distribution with a class of Gaussian process priors, we study the statistical performance of variational Bayesian inference to the posterior with variational sets consisting of Gaussian measures or a mean-field family. We propose certain conditions on the forward map $\mathcal{G}$, the variational set $\mathcal{Q}$ and the prior such that, as the number $N$ of measurements increases, the resulting variational posterior distributions contract to the ground truth $f_0$ generating the data, and derive a convergence rate with polynomial order or logarithmic order. As specific examples, we consider a collection of non-linear inverse problems, including the Darcy flow problem, the inverse potential problem for a subdiffusion equation, and the inverse medium scattering problem. Besides, we show that our convergence rates are minimax optimal for these inverse problems.

翻译：我们研究确定参数$f$的非线性贝叶斯反问题。针对一类高斯过程先验下的后验分布，我们研究了变分贝叶斯推断相对于由高斯测度或平均场族构成的变分集的后验统计性能。我们提出了前向映射$\mathcal{G}$、变分集$\mathcal{Q}$与先验分布的若干条件，使得当测量次数$N$增加时，所得的变分后验分布会收缩至生成数据的真实参数$f_0$，并推导出具有多项式阶或对数阶的收敛速率。作为具体实例，我们考察了一系列非线性反问题，包括达西渗流问题、次扩散方程的反势问题以及反介质散射问题。此外，我们证明了对于这些反问题，所得收敛速率是极小极大最优的。