In 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，深度学习技术在与逆问题相关的各项任务中取得了巨大成功，在这些逆问题中，感兴趣的目标量只能通过前向算子的间接测量进行观测。常见的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获得的重建结果。然而，后者方法计算成本高昂，且会引入测量数据中不存在的伪影，进而可能降低给定任务的性能。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求解所需任务的等变神经网络。为此，我们通过开发对前向算子域中已知对称性所诱导的数据变换具有等变性的网络层，构建了适当的网络结构。我们严格分析了测量算子与由此产生的群表示之间的关系，并证明了一个表征在给定群作用之间进行转换的线性算子类的表示定理。基于这一理论，我们将现有的李群等变深度学习概念扩展到逆问题，并引入由涉及的测量运算产生的新表示。这使得我们能够基于间接测量高效求解分类、回归乃至重建任务，即便在经典重建方法可能困难甚至不可行的极稀疏数据问题中同样适用。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。