In this work, we describe a new approach that uses variational encoder-decoder (VED) networks for efficient goal-oriented uncertainty quantification for inverse problems. Contrary to standard inverse problems, these approaches are \emph{goal-oriented} in that the goal is to estimate some quantities of interest (QoI) that are functions of the solution of an inverse problem, rather than the solution itself. Moreover, we are interested in computing uncertainty metrics associated with the QoI, thus utilizing a Bayesian approach for inverse problems that incorporates the prediction operator and techniques for exploring the posterior. This may be particularly challenging, especially for nonlinear, possibly unknown, operators and nonstandard prior assumptions. We harness recent advances in machine learning, i.e., VED networks, to describe a data-driven approach to large-scale inverse problems. This enables a real-time goal-oriented uncertainty quantification for the QoI. One of the advantages of our approach is that we avoid the need to solve challenging inversion problems by training a network to approximate the mapping from observations to QoI. Another main benefit is that we enable uncertainty quantification for the QoI by leveraging probability distributions in the latent space. This allows us to efficiently generate QoI samples and circumvent complicated or even unknown forward models and prediction operators. Numerical results from medical tomography reconstruction and nonlinear hydraulic tomography demonstrate the potential and broad applicability of the approach.

翻译：本文描述了一种新方法，利用变分编码器-解码器（VED）网络对反问题进行高效的目标导向不确定性量化。与传统反问题不同，这些方法具有“目标导向”特性：其目标是估计作为反问题解的函数的若干感兴趣量（QoI），而非解本身。此外，我们关注与QoI相关的不确定性度量计算，因此采用贝叶斯反问题框架，融合预测算子与后验探索技术。对于非线性（甚至未知）算子及非标准先验假设，这一过程可能尤为困难。我们借助机器学习领域的最新进展（即VED网络），提出一种面向大规模反问题的数据驱动方法，从而实现对QoI的实时目标导向不确定性量化。本方法的优势之一在于：通过训练网络逼近从观测数据到QoI的映射，避免了求解复杂反问题的需要。另一主要优势在于：利用潜在空间中的概率分布实现QoI的不确定性量化，从而高效生成QoI样本，并规避复杂甚至未知的正向模型与预测算子。医学层析成像重建与非线性水文层析成像的数值结果验证了该方法的潜力与广泛适用性。