DeepGreen: deep learning of Green's functions for nonlinear boundary value problems.

Area of Science:

• Applied Mathematics
• Computational Physics
• Deep Learning

Background:

• Boundary value problems (BVPs) are crucial in analyzing physical systems across engineering disciplines.
• Green's functions are effective for linear BVPs but not for nonlinear ones due to the loss of linear superposition.
• Solving nonlinear BVPs often requires iterative methods and initial guesses, limiting efficiency.

Purpose of the Study:

• To develop a novel deep learning approach for solving nonlinear boundary value problems.
• To overcome the limitations of traditional methods in handling nonlinear systems.
• To create a flexible tool for identifying fundamental solutions to complex nonlinear problems.

Main Methods:

• A dual-autoencoder deep learning architecture is proposed.
• The autoencoders learn an invertible coordinate transform to linearize the nonlinear BVP.
• The method identifies a linear operator and Green's function for solving new nonlinear BVPs.

Main Results:

• The deep learning method successfully solves various nonlinear BVPs, including Helmholtz, Sturm-Liouville, elasticity, and Poisson equations.
• The approach achieves solutions orders of magnitude faster than traditional methods.
• No initial guess is required for the nonlinear BVP solutions.

Conclusions:

• The proposed deep learning method offers a flexible and efficient solution for nonlinear boundary value problems.
• This approach combines the power of deep learning with the principles of Green's functions.
• It provides a significant advancement for computational physics and engineering applications.