Learning-based methods have demonstrated remarkable performance in solving inverse problems, particularly in image reconstruction tasks. Despite their success, these approaches often lack theoretical guarantees, which are crucial in sensitive applications such as medical imaging. Recent works by Arndt et al (2023 Inverse Problems 39 125018, 2024 Inverse Problems 40 045021) addressed this gap by analyzing a data-driven reconstruction method based on invertible residual networks (iResNets). They revealed that, under reasonable assumptions, this approach constitutes a convergent regularization scheme. However, the performance of the reconstruction method was only validated on academic toy problems and small-scale iResNet architectures. In this work, we address this gap by evaluating the performance of iResNets on two real-world imaging tasks: a linear blurring operator and a nonlinear diffusion operator. To do so, we extend some of the theoretical results from Arndt et al to encompass nonlinear inverse problems and offer insights for the design of large-scale performant iResNet architectures. Through numerical experiments, we compare the performance of our iResNet models against state-of-the-art neural networks, confirming their efficacy. Additionally, we numerically investigate the theoretical guarantees of this approach and demonstrate how the invertibility of the network enables a deeper analysis of the learned forward operator and its learned regularization.

翻译：基于学习的方法在解决逆问题，特别是图像重建任务中已展现出卓越性能。尽管这些方法取得了成功，但它们通常缺乏理论保证，而这在医学成像等敏感应用中至关重要。Arndt等人（2023 Inverse Problems 39 125018, 2024 Inverse Problems 40 045021）近期的工作通过分析一种基于可逆残差网络（iResNets）的数据驱动重建方法，弥补了这一空白。他们揭示出，在合理假设下，该方法构成了一种收敛的正则化方案。然而，该重建方法的性能仅在学术性示例问题和小规模iResNet架构上得到验证。在本工作中，我们通过评估iResNets在两个实际成像任务——线性模糊算子与非线性扩散算子——上的性能来填补这一空白。为此，我们将Arndt等人的部分理论结果扩展至非线性逆问题，并为设计大规模高性能iResNet架构提供了见解。通过数值实验，我们将所提出的iResNet模型与最先进的神经网络进行性能比较，证实了其有效性。此外，我们对该方法的理论保证进行了数值研究，并展示了网络的可逆性如何实现对所学前向算子及其所学正则化的更深入分析。