It is well known that Newton's method can have trouble converging if the initial guess is too far from the solution. Such a problem particularly occurs when this method is used to solve nonlinear elliptic partial differential equations (PDEs) discretized via finite differences. This work focuses on accelerating Newton's method convergence in this context. We seek to construct a mapping from the parameters of the nonlinear PDE to an approximation of its discrete solution, independently of the mesh resolution. This approximation is then used as an initial guess for Newton's method. To achieve these objectives, we elect to use a Fourier neural operator (FNO). The loss function is the sum of a data term (i.e., the comparison between known solutions and outputs of the FNO) and a physical term (i.e., the residual of 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 and anisotropic problems, with larger gains on coarse grids.

翻译：众所周知，若初始猜测离解过远，牛顿法可能难以收敛。当该方法用于求解通过有限差分离散化的非线性椭圆型偏微分方程时，此问题尤为突出。本工作聚焦于在此背景下加速牛顿法的收敛。我们旨在构建一个从非线性偏微分方程的参数到其离散解近似值的映射，该映射独立于网格分辨率。随后，此近似解被用作牛顿法的初始猜测。为实现这些目标，我们选择使用傅里叶神经算子。损失函数由数据项（即已知解与FNO输出之间的比较）和物理项（即偏微分方程离散化的残差）之和构成。一维和二维的数值结果表明，与朴素的初始猜测相比，所提出的初始猜测极大地加速了牛顿法的收敛，尤其对于高度非线性和各向异性问题，且在粗网格上收益更大。