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：物理引导的深度学习是科学机器学习中一个重要且普遍的研究主题，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，数据获取成本高昂，且决策需要高精度。本文针对非线性逆问题的变分建模，提出了一种高效的物理引导深度学习框架，并将其应用于求解电阻抗断层成像（EIT）逆问题。该框架通过将所提出的Anderson加速高斯-牛顿（GNAA）算法展开为端到端的深度学习方法来实现。首先，我们证明了GNAA算法在Anderson深度等于一和大于一两种情况下的收敛性。随后，我们结合GNAA与深度学习的互补优势，提出了三种策略：学习正则化的GNAA（GNAA-LRNet），其中正则化矩阵的奇异值通过深度神经网络学习；学习邻近算子的GNAA（GNAA-LPNet），其中正则化邻近算子通过深度神经网络学习；以及即插即用方法的GNAA（GNAA-PnPNet），其中正则化邻近算子由预训练的深度去噪器替代。最后，我们通过数值实验表明，所提出的方法显著提高了收敛速度和逆解的质量。