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：本文提出一种新方法，利用变分编码器-解码器（VED）网络实现反问题中高效的目标导向不确定性量化。与传统反问题不同，这类方法具有"目标导向性"：其目标是估计反问题解的某些感兴趣量（QoI），而非直接求解反问题本身。此外，我们致力于计算与QoI相关的不确定性指标，因此采用贝叶斯反问题框架，结合预测算子与后验探索技术。这对非线性（可能未知）算子及非标准先验假设尤其具有挑战性。我们借助机器学习最新进展（即VED网络），提出一种面向大规模反问题的数据驱动方法，实现对QoI的实时目标导向不确定性量化。本方法优势之一在于：通过训练网络逼近观测数据到QoI的映射，避免了求解困难的反问题。另一核心优势是：利用潜在空间概率分布实现对QoI的不确定性量化，从而高效生成QoI样本，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力层析成像的数值结果验证了该方法的潜力与广泛适用性。