Data-driven methods for the solution of inverse problems have become widely popular in recent years thanks to the rise of machine learning techniques. A popular approach concerns the training of a generative model on additional data to learn a bespoke prior for the problem at hand. In this article we present an analysis for such problems by presenting quantitative error bounds for minimum Wasserstein-2 generative models for the prior. We show that under some assumptions, the error in the posterior due to the generative prior will inherit the same rate as the prior with respect to the Wasserstein-1 distance. We further present numerical experiments that verify that aspects of our error analysis manifests in some benchmarks followed by an elliptic PDE inverse problem where a generative prior is used to model a non-stationary field.

翻译：近年来，得益于机器学习技术的兴起，数据驱动的反问题求解方法已得到广泛普及。一种主流方法涉及通过额外数据训练生成模型，从而为特定问题学习定制化的先验分布。本文针对此类问题展开分析，通过建立基于最小Wasserstein-2距离生成先验模型的定量误差界，证明在某些假设条件下，由生成先验引起的后验误差将继承先验分布关于Wasserstein-1距离的收敛速率。我们进一步通过数值实验验证误差分析的若干特性：首先在基准测试中展示理论结果，随后将其应用于椭圆型偏微分方程反问题案例，其中生成先验被用于建模非平稳随机场。