Core Concepts
Fourier neural operators accelerate Newton's method convergence for nonlinear elliptic PDEs.
Abstract
The content discusses using Fourier neural operators to improve the convergence of Newton's method for nonlinear elliptic PDEs. It provides insights into the training process, loss functions, and numerical results for different nonlinearity levels.
- Introduction to Newton's method for nonlinear systems.
- Overview of Fourier neural operators and their application.
- Training strategies with different loss functions.
- Comparison of predicted initial guesses with naive initial guesses for different nonlinearity levels.
- Analysis of gains in number of iterations and CPU time for different mesh resolutions.
- Results for α0 = 2, α0 = 5, and α0 = 8 cases.
Stats
Newton's method는 초기 추측이 해결책과 멀리 떨어져 있을 때 수렴에 어려움을 겪을 수 있습니다.
Fourier neural operators는 Newton's method의 수렴을 가속화하는 데 효과적입니다.
Quotes
"Newton's method is often used for smaller problems."
"Fourier neural operators provide a good approximation of the solution."