This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.

翻译：本文研究高斯代理模型在线性偏微分方程贝叶斯反问题中的应用，特别关注训练数据量较少的情况。在此情形下，高斯先验的类型对代理模型在贝叶斯反演中的性能至关重要。我们扩展了Raissi等人(2017)的框架，构建了基于偏微分方程信息的高斯先验，并利用这些先验构建不同的近似后验。多项数值实验表明，基于偏微分方程信息的高斯先验优于传统先验。