DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems

Abstract

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.

Data-driven discovery of Green's functions with human-understandable deep learning

“…where L denotes a linear operator, N is a nonlinear operator, and < 1 is a small parameter controlling the nonlinearity. We demonstrate this ability on the three nonlinear boundary value problems, dominated by the linearity, used in (8). [−p(x)u ] + q(x)(u + u 3 ) = f (x), u(0) = u(2π) = 0, with p(x) = 0.4 sin(x) − 3, q(x) = 0.6 sin(x) − 2, and = 0.4.…”

“…Finally, we emphasize that the enhanced approximation properties of rational NNs (15) make them ideal for learning Green's functions and, more generally, approximating functions within regression problems. These networks may also be of benefit to other approaches for solving and learning PDEs with DL techniques, such as PINNs (37), DeepGreen (8), DeepONet (7),…”

“…of nature. Recently, scientific computing and machine learning have successfully converged on PDE discovery (3)(4)(5), PDE learning (6)(7)(8)(9)(10), and symbolic regression (11,12) as promising means for applying machine learning to scientific investigations. These PDE learning techniques attempt to discover the coefficients of a PDE model or learn the operator that maps excitations to system responses.…”

“…In Fig. 4A to C, we visualize the Green's function NNs of three operators with cubic nonlinearity considered in (8). The nonlinearity does not prevent our method from discovering the Green's function of an underlying linearized model, from which one can understand features such as symmetry and boundary conditions.…”

Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows

“…Recent developments in deep learning for engineering problems bring advanced and innovative approaches to improve the efficiency, flexibility, and accuracy of the predictive models. Some of the outstanding applications of deep neural networks (DNNs) in the domain of computational physics are solution of partial differential equations (PDEs) (Raissi et al, 2019), operator learning (Gin et al, 2020;Lu et al, 2021;Li et al, 2021), linear embedding of nonlinear dynamics (Lusch et al, 2018), and model reduction of dynamical systems (Lee and Carlberg, 2020). Fluid mechanics has been one of the active research topics for development of innovative DNN-based approaches (Kutz, 2017;Duraisamy et al, 2019;Brunton et al, 2020).…”