针对非线性方程组中非线性不平衡导致 Newton 求解困难的问题，提出残差驱动的自适应权重乘子策略，以低开销改善步长选择、效率与鲁棒性。
摘要：
Newton 型求解器已被广泛用于求解各类非线性代数方程组。然而，对于某些复杂的非线性代数方程组，高效求解这些系统仍然是一项具有挑战性的任务。造成这一挑战的主要原因是非线性系统内部不平衡的非线性。因此，准确识别并平衡系统中不平衡的非线性至关重要。在本工作中，我们提出一种残差驱动的自适应策略，用于识别并平衡系统中的非线性。该策略背后的基本思想是为非线性系统的每个分量分配一个自适应权重乘子；随着残差分量增大，这些权重乘子会按照特定更新规则增加，从而使 Newton 型求解器能够选择更合适的步长，确保非线性系统中的每个分量都得到充分下降，而不是彼此竞争。更重要的是，我们的策略只产生可忽略的额外计算开销，并且能够与其他 Newton 型求解器无缝集成，有助于提升其效率和鲁棒性。我们在多种基准问题上测试了我们的算法，包括一个化学平衡系统、一个对流扩散问题以及一系列具有挑战性的非线性系统。实验结果表明，我们的算法不仅在计算效率方面优于现有 Newton 型求解器，而且表现出更强的鲁棒性，尤其是在处理具有高度不平衡非线性的系统时。
英文摘要：
Newton-type solvers have been extensively employed for solving a variety of nonlinear system of algebraic equations. However, for some complex nonlinear system of algebraic equations, efficiently solving these systems remains a challenging task. The primary reason for this challenge arises from the unbalanced nonlinearities within the nonlinear system. Therefore, accurately identifying and balancing the unbalanced nonlinearities in the system is essential. In this work, we propose a residual-driven adaptive strategy to identify and balance the nonlinearities in the system. The fundamental idea behind this strategy is to assign an adaptive weight multiplier to each component of the nonlinear system, with these weight multipliers increasing according to a specific update rule as the residual components increase, thereby enabling the Newton-type solver to select a more appropriate step length, ensuring that each component in the nonlinear system experiences sufficient reduction rather than competing against each other. More importantly, our strategy yields negligible additional computational overhead and can be seamlessly integrated with other Newton-type solvers, contributing to the improvement of their efficiency and robustness. We test our algorithm on a variety of benchmark problems, including a chemical equilibrium system, a convective diffusion problem, and a series of challenging nonlinear systems. The experimental results demonstrate that our algorithm not only outperforms existing Newton-type solvers in terms of computational efficiency but also exhibits superior robustness, particularly in handling systems with highly imbalanced nonlinearities.
A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics
organization= Department of Mathematics , addressline= The Hong Kong University of Science；organization= Mathematical Institute , addressline= University of Oxford , country= United Kingdom；organization= Laboratoire Jacques-Louis Lions (UMR 7598) , addressline= Sorbonne Université , country= France；Autonomous Driving Research Lab , addressline= HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute , city= Futian , state= Shenzhen , country= China
Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method
University of Strathclyde（斯特拉斯克莱德大学）；Université de Genève（日内瓦大学）；Université Nice Sophia Antipolis（尼斯索菲亚安蒂波利斯大学）；Inria（法国国家数字科学研究所）；Hong Kong Baptist University（香港浸会大学）
A space-time LATIN-PGD strategy for solving Newtonian compressible flows
Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS - Laboratoire de Mécanique Paris-Saclay（巴黎萨克雷大学，中央理工-苏佩莱克学院，巴黎萨克雷高等师范学院，法国国家科学研究中心，巴黎萨克雷力学实验室）；Leibniz University Hanover, Institute of Continuum Mechanics（汉诺威莱布尼茨大学，连续介质力学研究所）
Application of polynomial algebras to non-linear equation solvers
Insitut Universitari de Matemàtiques i Aplicacions de Castelló（卡斯特利翁数学与应用数学研究所）；Universitat Jaume I（海梅一世大学）；Univesitat Politècnica de Catalunya - Barcelona Tech（加泰罗尼亚理工大学-巴塞罗那技术学院）
Learning features from Newton's algorithm: a way to accelerate nonlinear parametrized PDE solvers
Centre Borelli（博雷利中心）；ENS Paris-Saclay（巴黎萨克雷高等师范学校）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Michelin（米其林公司）；Centre de Recherche de Ladoux（拉杜研究中心）；BMBI（BMBI中心）；Université de Technologie de Compiègne（贡比涅技术大学）
A TFETI Domain Decomposition Solver for Elastoplastic Problems
VŠB-Technical University of Ostrava（俄斯特拉发理工大学）；Institute of Geonics AS CR, v.v.i.（捷克科学院地质力学研究所）；The Institute of Information Theory and Automation（信息与自动化研究所）
提出基于 TFETI 区域分解方法的并行算法，结合半光滑牛顿法和隐式欧拉时间离散，高效求解具有线性各向同性硬化 von Mises 准则的弹塑性问题。
BandPC: Learning Residual-Band Preconditioner Combinations for Flexible Conjugate Gradient Solvers
Wuhan University of Technology（武汉理工大学）；Embodied Physical Intelligence Center in Civil Engineering, Wuhan University of Technology（武汉理工大学土木工程具身智能中心）；Key Laboratory of Low-Altitude Technology and Smart Urban Renewal, Department of Housing and Urban-Rural Development of Hubei Province, Wuhan University of Technology（湖北省住房和城乡建设厅低空技术与智慧城市更新重点实验室（武汉理工大学））；Sanya Science and Education Innovation Park of Wuhan University of Technology（武汉理工大学三亚科教创新园）
