We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled by physical laws but is specified via training pairs (consisting of images and data) of the input-output relation of the operator. We derive a stable method for computing the operator, which consists of first a Gram-Schmidt orthonormalization of images and a principal component analysis of the data. This two-step algorithm provides a spectral decomposition of the linear operator. Moreover, we show that both Gram-Schmidt and principal component analysis can be written as a deep neural network, which relates this procedure to de-and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly exploiting the physical model.

翻译：本文研究求解可能病态的线性算子方程，其中算子并非由物理定律建模，而是通过算子输入输出关系的训练对（由图像和数据组成）来指定。我们推导了一种稳定的算子计算方法，该方法首先对图像进行Gram-Schmidt正交归一化处理，再对数据进行主成分分析。这一两步算法提供了线性算子的谱分解。此外，我们证明Gram-Schmidt过程与主成分分析均可表示为深度神经网络，从而将该流程与编解码器网络相关联。因此，我们将该两步算法称为线性代数网络。最后，通过数值模拟验证了该策略在重构谱函数及求解算子方程方面的可行性，且无需显式利用物理模型。