It is well known that Newton's method, especially when applied to large problems such as the discretization of nonlinear partial differential equations (PDEs), can have trouble converging if the initial guess is too far from the solution. This work focuses on accelerating this convergence, in the context of the discretization of nonlinear elliptic PDEs. We first provide a quick review of existing methods, and justify our choice of learning an initial guess with a Fourier neural operator (FNO). This choice was motivated by the mesh-independence of such operators, whose training and evaluation can be performed on grids with different resolutions. The FNO is trained using a loss minimization over generated data, loss functions based on the PDE discretization. Numerical results, in one and two dimensions, show that the proposed initial guess accelerates the convergence of Newton's method by a large margin compared to a naive initial guess, especially for highly nonlinear or anisotropic problems.

翻译：众所周知，牛顿法在应用于大规模问题（如非线性偏微分方程离散化）时，若初始猜测远离解，可能难以收敛。本文聚焦于非线性椭圆偏微分方程离散化背景下加速这一收敛过程。我们首先简要回顾现有方法，并论证选择利用傅里叶神经算子学习初始猜测的合理性。该选择源于此类算子的网格无关性——其训练与评估可在不同分辨率的网格上进行。傅里叶神经算子通过基于偏微分方程离散化的损失函数，在生成数据上最小化损失进行训练。一维与二维数值结果表明，相较于朴素初始猜测，本文提出的初始猜测能显著加速牛顿法收敛，尤其适用于强非线性或各向异性问题。