In the recent years, deep learning techniques have shown great success in various tasks related to inverse problems, where a target quantity of interest can only be observed through indirect measurements by a forward operator. Common approaches apply deep neural networks in a post-processing step to the reconstructions obtained by classical reconstruction methods. However, the latter methods can be computationally expensive and introduce artifacts that are not present in the measured data and, in turn, can deteriorate the performance on the given task. To overcome these limitations, we propose a class of equivariant neural networks that can be directly applied to the measurements to solve the desired task. To this end, we build appropriate network structures by developing layers that are equivariant with respect to data transformations induced by well-known symmetries in the domain of the forward operator. We rigorously analyze the relation between the measurement operator and the resulting group representations and prove a representer theorem that characterizes the class of linear operators that translate between a given pair of group actions. Based on this theory, we extend the existing concepts of Lie group equivariant deep learning to inverse problems and introduce new representations that result from the involved measurement operations. This allows us to efficiently solve classification, regression or even reconstruction tasks based on indirect measurements also for very sparse data problems, where a classical reconstruction-based approach may be hard or even impossible. We illustrate the effectiveness of our approach in numerical experiments and compare with existing methods.

翻译：近年来，深度学习技术在与反问题相关的各项任务中取得了巨大成功，其中目标感兴趣量只能通过前向算子进行间接测量来观测。常见方法是将深度神经网络作为后处理步骤应用于经典重建方法所获得的图像重建结果。然而，经典方法计算成本高，且会引入测量数据中不存在的伪影，进而可能降低在特定任务上的性能。为克服这些局限，我们提出了一类可直接应用于测量数据以解决所需任务的等变神经网络。为此，我们通过开发对前向算子领域中已知对称性所诱导的数据变换具有等变性的网络层，构建了相应的网络结构。我们严格分析了测量算子与所得群表示之间的关系，并证明了表征定理，该定理刻画了在给定群作用之间进行变换的线性算子类别。基于这一理论，我们将李群等变深度学习的现有概念扩展至反问题领域，并引入了由涉及的测量操作所产生的新表示。这使得我们能够基于间接测量高效解决分类、回归甚至重建任务，尤其适用于经典基于重建方法难以甚至无法处理的稀疏数据问题。我们通过数值实验验证了该方法的效果，并与现有方法进行了比较。