Deep Learning Solution of the Eigenvalue Problem for Differential Operators.

Area of Science:

• Numerical Analysis
• Scientific Computing
• Machine Learning

Background:

• Classical numerical methods for eigenvalue problems often involve complex discretization and produce non-analytic approximations.
• Solving eigenvalue problems for differential operators is crucial across various scientific disciplines.

Purpose of the Study:

• To introduce a novel, unsupervised neural network-based solver for eigenvalue problems of differential self-adjoint operators.
• To develop training procedures for increasingly complex eigenvalue problems and demonstrate the solver's capability to find multiple eigenpairs.
• To analyze the numerical error of the proposed neural network method.

Main Methods:

• A neural network is employed to learn eigenpairs in an unsupervised, end-to-end manner.
• Training procedures are proposed to handle progressively challenging eigenvalue problems.
• The method is applied to the Laplacian operator and the Legendre differential equation on various domains, including free-form ones.

Main Results:

• The neural network solver successfully finds multiple smallest eigenpairs for differential operators.
• Demonstrated efficacy on the Laplacian operator (relevant to image processing, computer vision, shape analysis) and Legendre differential equation.
• The method is applicable to free-form domains like L-shape and circular cut domains.
• An analysis provides an upper bound for the solution error based on truncation error and network structure.

Conclusions:

• The proposed neural network solver offers a powerful, flexible alternative to classical methods for eigenvalue problems.
• The unsupervised, end-to-end approach simplifies the solution process and handles complex domains effectively.
• The error analysis provides valuable insights into the method's reliability and performance.