Variational Green's Functions for Volumetric PDEs
This addresses a bottleneck in tasks like shape analysis and physical simulation for researchers and practitioners in computational physics and geometry processing, though it is incremental as it builds on existing neural field and variational methods.
The paper tackles the computational challenge of evaluating Green's functions for linear self-adjoint PDEs on arbitrary geometries by introducing Variational Green's Function (VGF), which learns a smooth, differentiable representation that decomposes into an analytic free-space component and a learned corrector, resulting in fast evaluation and differentiability.
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.