In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an \textit{a-priori} stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by discussing an example that fulfills assumptions of the paper.

翻译：本研究提出并分析了一种随机梯度下降法的新变体，称为随机数据驱动Bouligand-Landweber迭代法，专门用于求解非光滑不适定反问题系统。该方法通过有界线性算子利用训练数据引导迭代过程。在每次迭代中，方法从带有数据驱动项的非线性系统中随机选择一个方程。针对精确数据，已证明均方迭代误差收敛至零。而当数据包含噪声时，我们采用该方法结合预设停准则（称为先验停止规则）。我们在无限维希尔伯特空间框架下建立了该方案收敛性与稳定性的完整理论基础，并通过满足论文假设条件的算例进一步佐证了相关理论依据。