Physics-guided deep learning is an important prevalent research topic in scientific machine learning, which has tremendous potential in various complex applications including science and engineering. In these applications, data is expensive to acquire and high accuracy is required for making decisions. In this work, we introduce an efficient physics-guided deep learning framework for the variational modeling of nonlinear inverse problems, which is then applied to solve an electrical impedance tomography (EIT) inverse problem. The framework is achieved by unrolling the proposed Anderson accelerated Gauss-Newton (GNAA) algorithm into an end-to-end deep learning method. Firstly, we show the convergence of the GNAA algorithm in both cases: Anderson depth is equal to one and Anderson depth is greater than one. Then, we propose three types of strategies by combining the complementary strengths of GNAA and deep learning: GNAA of learned regularization (GNAA-LRNet), where the singular values of the regularization matrix are learned by a deep neural network; GNAA of learned proximity (GNAA-LPNet), where the regularization proximal operator is learned by using a deep neural network; GNAA of plug-and-play method (GNAA-PnPNet) where the regularization proximal operator is replaced by a pre-trained deep denoisers. Lastly, we present some numerical experiments to illustrate that the proposed approaches greatly improve the convergence rate and the quality of inverse solutions.

翻译：物理引导深度学习是科学机器学习中重要的前沿研究方向，在科学与工程等各类复杂应用中具有巨大潜力。在这些应用中，数据采集成本高昂，且决策过程对精度要求极高。本文提出了一种高效的物理引导深度学习框架，用于非线性反问题的变分建模，并将其应用于电阻抗成像（EIT）反问题的求解。该框架通过将所提出的安德森加速高斯-牛顿（GNAA）算法展开为端到端深度学习算法来实现。首先，我们证明了GNAA算法在安德森深度等于1和大于1两种情况下的收敛性。随后，通过结合GNAA与深度学习的互补优势，提出三种策略：GNAA学习正则化网络（GNAA-LRNet），利用深度神经网络学习正则化矩阵的奇异值；GNAA学习邻近算子网络（GNAA-LPNet），利用深度神经网络学习正则化邻近算子；GNAA即插即用网络（GNAA-PnPNet），将正则化邻近算子替换为预训练的深度去噪器。最后，通过数值实验表明，所提方法能够显著提升收敛速度与反问题解的质量。