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.

从数据中学习先验模型，使用生成模型以高效学习先验的推进映射，并结合神经算子逼近方法并行学习前向模型，为需要从间接数据学习先验的情况提供了计算上更有效的方法。