  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.
