DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems
This work provides a novel approach for solving nonlinear boundary value problems, which are prevalent in many engineering and physics domains, by extending the utility of Green's functions to previously intractable nonlinear systems.
This paper introduces DeepGreen, a deep learning method that solves nonlinear boundary value problems (BVPs) by learning an invertible coordinate transform to linearize the BVP. This allows for the identification of a linear operator and a Green's function, which can then be used to solve new nonlinear BVPs across various systems like nonlinear Helmholtz, Sturm-Liouville, elasticity, and 2D nonlinear Poisson equations.
Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, quantum mechanics, and elasticity. The fundamental solution, or Green's function, is a leading method for solving linear BVPs that enables facile computation of new solutions to systems under any external forcing. However, fundamental Green's function solutions for nonlinear BVPs are not feasible since linear superposition no longer holds. In this work, we propose a flexible deep learning approach to solve nonlinear BVPs using a dual-autoencoder architecture. The autoencoders discover an invertible coordinate transform that linearizes the nonlinear BVP and identifies both a linear operator $L$ and Green's function $G$ which can be used to solve new nonlinear BVPs. We find that the method succeeds on a variety of nonlinear systems including nonlinear Helmholtz and Sturm--Liouville problems, nonlinear elasticity, and a 2D nonlinear Poisson equation. The method merges the strengths of the universal approximation capabilities of deep learning with the physics knowledge of Green's functions to yield a flexible tool for identifying fundamental solutions to a variety of nonlinear systems.