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，用于获取基于偏微分方程逆问题的概率解。贝叶斯方法因具备不确定性量化能力而受到青睐，但这类方法通常假设似然模型或数据生成过程已知。然而在许多逆问题中，这一假设难以成立——数据生成过程的明确规范往往并不直观。我们采用吉布斯后验框架，直接在参数的概率分布空间上构建正则化变分问题。我们提出一种新颖的模型比较框架，通过评估特定损失函数的"预测性能"来判断其最优性。我们提供了交叉验证程序，用于校准变分目标中的正则化参数并比较多种损失函数。本文还展示了吉布斯后验的一些全新理论性质。通过一个以超声振动测量中用于表征动脉血管的色散波动模型为背景的模拟案例，我们验证了该框架的实用性。