  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.
