发表机构
Centre Borelli; ENS Paris-Saclay; Université Paris-Saclay; CNRS; Michelin; Centre de Recherche de Ladoux; BMBI; Université de Technologie de Compiègne（博雷利中心; 巴黎萨克雷高等师范学校; 巴黎萨克雷大学; 法国国家科学研究中心; 米其林公司; 拉杜研究中心; BMBI中心; 贡比涅技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出两阶段牛顿初始猜测策略，通过学习特征构建降维空间，结合回归模型与GMRES校正，减少牛顿迭代次数与CPU时间，加速非线性参数化PDE求解器。
AI 中文摘要
众所周知，当初始猜测值接近非线性方程组的根时，牛顿法的收敛速度更快。本文提出了一种两阶段牛顿初始猜测策略，该策略通过从参数空间采样和预计算解的数据库中学习特征来构建。该方法利用离散牛顿轨迹构造两个互补的降维空间：一个是由收敛状态构建的解特征空间，另一个是由中间牛顿增量构建的校正搜索方向特征空间。对于未见过的参数，使用回归模型预测代理解近似值。然后，在第二步中，使用专用的基于GMRES的方法计算最小化残差的校正项。得到的状态随后作为高保真牛顿法的初始猜测，完成收敛。由于该校正步骤仅需要残差评估和小型最小二乘问题的求解，因此计算成本较低。一旦有高保真残差场和基于脚本的编程接口，该方法就具有弱侵入性。该策略减少了牛顿迭代次数并降低了总体CPU时间。对具有代表性的偏微分方程问题进行的数值实验表明，与单独的代理初始化相比，可实现量化的加速，且加速效果显著。这种通用方法可应用于广泛的大规模非线性问题。
英文摘要
It is well known that Newton's method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.
