Variational Green's Functions for Volumetric PDEs

TL;DR

This paper introduces Variational Green's Function (VGF), a neural method to efficiently learn and evaluate Green's functions for linear PDEs on arbitrary geometries, addressing computational challenges and boundary condition enforcement.

Contribution

The paper presents a novel variational approach that decomposes Green's functions into analytic and learned parts, enabling fast, differentiable evaluation with boundary condition handling.

Findings

VGF accurately models Green's functions for key PDEs.

The learned functions are fast to evaluate and differentiable.

The method effectively incorporates boundary conditions.

Abstract

Green's functions characterize the fundamental solutions of partial differential equations; they are essential for tasks ranging from shape analysis to physical simulation, yet they remain computationally prohibitive to evaluate on arbitrary geometric discretizations. We present Variational Green's Function (VGF), a method that learns a smooth, differentiable representation of the Green's function for linear self-adjoint PDE operators, including the Poisson, the screened Poisson, and the biharmonic equations. To resolve the sharp singularities characteristic of the Green's functions, our method decomposes the Green's function into an analytic free-space component, and a learned corrector component. Our method leverages a variational foundation to impose Neumann boundary conditions naturally, and imposes Dirichlet boundary conditions via a projective layer on the output of the neural field. The resulting Green's functions are fast to evaluate, differentiable with respect to source application, and can be conditioned on other signals parameterizing our geometry.