发表机构
Delft University of Technology; Deltares（代尔夫特理工大学; 三角洲研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文综述了机器学习增强的混合迭代方法，用于求解线性和非线性方程组，在保持经典方法可解释性的同时提升效率，并讨论了开放挑战与未来方向。
AI 中文摘要
方程组在广泛的科学和工程应用中产生。本文聚焦于一般方程组的求解器，包括但不限于由偏微分方程产生的方程组。这些系统可大致分为线性和非线性问题。对于大型线性系统，迭代求解器通常优于直接方法，因为直接方法的计算成本呈超线性增长。尽管在系数矩阵的某些假设下收敛理论已发展成熟，但许多类别的系统仍面临开放性挑战。对于非线性方程组，这些困难变得更加严重，因为非线性求解器通常依赖于反复线性化。例如，牛顿法在解附近可能二次收敛；但当初始猜测选择不当时，它也可能收敛缓慢或发散。存在具有多种变体和超参数设置的广泛求解器，开发高效且稳健的迭代方法仍是一个活跃的研究领域。近年来，机器学习（ML）技术已被应用于增强经典迭代方法的效率，同时保持其可解释性和可靠性。我们将这些ML增强的迭代方法称为混合迭代方法，因为它们将经典迭代方法与ML相结合。本文全面概述了为线性和非线性方程组构建混合迭代方法的最新方法，同时讨论了开放性挑战并概述了未来研究的潜在方向。
英文摘要
Systems of equations arise in a wide range of scientific and engineering applications. The present work focuses on solvers for general systems of equations, including but not limited to those arising from partial differential equations. These systems can be broadly categorized into linear and nonlinear problems. For large linear systems, iterative solvers are generally preferred over direct methods due to the latter's superlinear growth of computational costs. Although convergence theory is well-developed under certain assumptions on the coefficient matrix, many classes of systems still pose open challenges. These difficulties become even more severe for systems of nonlinear equations, where nonlinear solvers typically rely on repeated linearization. For example, Newton's method may even converge quadratically near the solution; it can also converge slowly or diverge when the initial guess is not chosen appropriately. A wide range of solvers with diverse variants and hyperparameter settings exists, and the development of efficient and robust iterative methods remains an active area of research. Recently, machine learning (ML) techniques have been applied to enhance the efficiency of classical iterative methods while preserving their interpretability and reliability. We refer to these ML-enhanced iterative methods as hybrid iterative methods, in the sense that they combine classical iterative methods with ML. This paper provides a comprehensive overview of state-of-the-art approaches to constructing hybrid iterative methods for systems of both linear and nonlinear equations, while also discussing open challenges and outlining potential directions for future research.
Comments78 pages
