Accelerating the convergence of Newton’s method for nonlinear elliptic PDEs using Fourier neural operators

• Published in: Communications in nonlinear science & numerical simulation 2025-01, Vol.140 (2), p.108434, Article 108434
• Main Authors: Aghili, Joubine, Franck, Emmanuel, Hild, Romain, Michel-Dansac, Victor, Vigon, Vincent
• Format: Article
• Language: English
• Subjects: Computer Science; Fourier neural operators; Machine Learning; Mathematics; Neural networks; Newton’s method; Nonlinear elliptic PDEs; Numerical Analysis
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2024.108434

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. •Introduce a Fourier Neural Operator to map parameters of a nonlinear elliptic PDE.•Approximate the discrete solution of the PDE using this mapping.•Train this FNO using a novel discretization-informed loss function.•This FNO provides an initial guess for Newton’s method used in solving the PDE.•These new initial guesses are better than naive ones, leading to faster convergence.