Bayesian inversion is central to the quantification of uncertainty within problems arising from numerous applications in science and engineering. To formulate the approach, four ingredients are required: a forward model mapping the unknown parameter to an element of a solution space, often the solution space for a differential equation; an observation operator mapping an element of the solution space to the data space; a noise model describing how noise pollutes the observations; and a prior model describing knowledge about the unknown parameter before the data is acquired. This paper is concerned with learning the prior model from data; in particular, learning the prior from multiple realizations of indirect data obtained through the noisy observation process. The prior is represented, using a generative model, as the pushforward of a Gaussian in a latent space; the pushforward map is learned by minimizing an appropriate loss function. A metric that is well-defined under empirical approximation is used to define the loss function for the pushforward map to make an implementable methodology. Furthermore, an efficient residual-based neural operator approximation of the forward model is proposed and it is shown that this may be learned concurrently with the pushforward map, using a bilevel optimization formulation of the problem; this use of neural operator approximation has the potential to make prior learning from indirect data more computationally efficient, especially when the observation process is expensive, non-smooth or not known. The ideas are illustrated with the Darcy flow inverse problem of finding permeability from piezometric head measurements.

翻译：贝叶斯反演是科学与工程众多应用领域中量化不确定性的核心方法。该方法构建需要四个要素：将未知参数映射到解空间（常为微分方程解空间）的正向模型；将解空间元素映射到数据空间的观测算子；描述观测值受噪声污染的噪声模型；以及在获取数据前描述未知参数先验知识的先验模型。本文研究从数据中学习先验模型的问题，特别是从通过含噪观测过程获得的间接数据多重实现中学习先验。先验模型通过生成模型表示为潜空间高斯分布的推前映射；该推前映射通过最小化适当损失函数进行学习。采用在经验逼近下良定义的度量来构建推前映射的损失函数，从而形成可实现的算法框架。进一步提出一种高效的基于残差的神经算子来逼近正向模型，并证明该算子可与推前映射通过问题的双层优化公式进行协同学习；这种神经算子逼近方法有望显著提升基于间接数据的先验学习计算效率，尤其适用于观测过程计算昂贵、非光滑或未知的情形。本文以达西流反问题——通过测压水头观测反演渗透率——为例阐释所提方法。