Variational Green's Functions for Volumetric PDEs

TL;DR

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.

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.

Figures (18)

• Figure 1: The analytic-corrector decomposition of the Neumann Laplace Green's function uses the analytic Green's function to capture the sharpness of the Green's function, letting our neural field capture an additional smooth corrector field required to enforce the 0-Neumann boundary conditions.
• Figure 2: Leveraging the sifting property of the Dirac delta function is essential for capturing the sharpness of the impulse source.
• Figure 3: Minimizing the Dirichlet energy naturally enforces vanishing Neumann boundary conditions in our Green's functions for the Laplace operator. These boundary conditions are preserved across various impulse sources all generated by the same neural field.
• Figure 4: Vanishing Neumann boundary conditions can be enforced naturally by our two-stage mixed method to solving for the Green's functions of the biharmonic operator. These boundary conditions are preserved across various queried impulse source locations, all generated by the same pre-trained neural field.
• Figure 5: The analytic corrector decomposition allows our final learned Green's function to properly capture the sharp singularity characteristic of the Laplacian's Green's function on a rectangular domain.