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。

间接数据的有效先验校准

Efficient Prior Calibration From Indirect Data

We study parameterisation-independent closed planar curve matching as a
Bayesian inverse problem. The motion of the curve is modelled via a curve on
the diffeomorphism group acting on the ambient space, leading to a large
deformation diffeomorphic metric mapping (LDDMM) functional penalising the
kinetic energy of the deformation. We solve Hamilton's equations for the curve
matching problem using the Wu-Xu element [S. Wu, J. Xu, Nonconforming finite
element spaces for $2m^\text{th}$ order partial differential equations on
$\mathbb{R}^n$ simplicial grids when $m=n+1$, Mathematics of Computation 88
(316) (2019) 531-551] which provides mesh-independent Lipschitz constants for
the forward motion of the curve, and solve the inverse problem for the momentum
using Bayesian inversion. Since this element is not affine-equivalent we
provide a pullback theory which expedites the implementation and efficiency of
the forward map. We adopt ensemble Kalman inversion using a negative Sobolev
norm mismatch penalty to measure the discrepancy between the target and the
ensemble mean shape. We provide several numerical examples to validate the
approach.

本研究采用 LDDMM functional 描述曲线的运动，在此基础上通过 Hamilton's equations 采用 Wu-Xu 元素求解曲线匹配问题，再利用贝叶斯反演解决动量问题，并采用集合 kalman 反演确认该方法可行。

使用贝叶斯反演进行平面曲线注册

Planar Curve Registration using Bayesian Inversion

Positron emission tomography (PET) is an important functional medical imaging
technique often used in the evaluation of certain brain disorders, whose
reconstruction problem is ill-posed. The vast majority of reconstruction
methods in PET imaging, both iterative and deep learning, return a single
estimate without quantifying the associated uncertainty. Due to ill-posedness
and noise, a single solution can be misleading or inaccurate. Thus, providing a
measure of uncertainty in PET image reconstruction can help medical
practitioners in making critical decisions. This paper proposes a deep
learning-based method for uncertainty quantification in PET image
reconstruction via posterior sampling. The method is based on training a
conditional generative adversarial network whose generator approximates
sampling from the posterior in Bayesian inversion. The generator is conditioned
on reconstruction from a low-dose PET scan obtained by a conventional
reconstruction method and a high-quality magnetic resonance image and learned
to estimate a corresponding standard-dose PET scan reconstruction. We show that
the proposed model generates high-quality posterior samples and yields
physically-meaningful uncertainty estimates.

本文提出了一种基于深度学习的方法，通过后验抽样在 PET 图像重建中进行不确定性量化。该方法基于条件生成对抗网络，其生成器近似于采样自贝叶斯反演中的后验。该文表明，所提出的模型生成高质量的后验样本，并产生具有物理意义的不确定性估计。