We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This assumption is difficult to justify in many inverse problems, where the specification of the data generation process is not obvious. We adopt a Gibbs posterior framework that directly posits a regularized variational problem on the space of probability distributions of the parameter. We propose a novel model comparison framework that evaluates the optimality of a given loss based on its "predictive performance". We provide cross-validation procedures to calibrate the regularization parameter of the variational objective and compare multiple loss functions. Some novel theoretical properties of Gibbs posteriors are also presented. We illustrate the utility of our framework via a simulated example, motivated by dispersion-based wave models used to characterize arterial vessels in ultrasound vibrometry.

翻译：我们提出一个通用框架，用于获得基于偏微分方程的逆问题的概率解。贝叶斯方法虽在不确定性量化方面具有吸引力，但其依赖于对似然模型或数据生成过程的先验知识。在许多逆问题中，这一假设难以成立，因为数据生成过程的设定并非显而易见。我们采用吉布斯后验框架，直接在参数概率分布空间上构建正则化变分问题。我们提出一种新的模型比较框架，通过损失函数的“预测表现”评估其最优性。我们提供交叉验证程序以校准变分目标中的正则化参数，并比较多个损失函数。本文还介绍了吉布斯后验的一些新理论性质。通过一个模拟实例（灵感来源于用于超声振动测量中动脉血管表征的色散波模型），我们展示了所提框架的实用性。