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.

中文翻译：

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使用傅里叶神经算子加速牛顿非线性椭圆偏微分方程方法的收敛


众所周知，如果初始估计值与解相差太远，牛顿方法可能会难以收敛。当这种方法用于求解通过有限差分离散化的非线性椭圆偏微分方程 （PDE） 时，尤其会出现这样的问题。这项工作的重点是在这种情况下加速牛顿方法的收敛。我们试图构建从非线性偏微分方程的参数到其离散解的近似值的映射，而与网格分辨率无关。然后，此近似值用作 Newton 方法的初始估计值。为了实现这些目标，我们选择使用傅里叶神经算子 （FNO）。损失函数是数据项（即 FNO 的已知解和输出之间的比较）和物理项（即 PDE 离散化的残差）之和。一维和二维的数值结果表明，与朴素的初始猜测相比，所提出的初始估计大大加快了牛顿方法的收敛速度，特别是对于高度非线性和各向异性问题，在粗网格上具有更大的增益。