Variational Green's Functions for Volumetric PDEs

2026-02-12

Technology Category

Machine Learning: Efficient ML / Green AIComputer Vision: Generative Adversarial Networks (GANs) for VisionSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructuresSearch and Retrieval-Augmented AI: Vertical and domain-specific search

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.

Problem

• Green's functions
• partial differential equations
• volumetric discretization
• computational efficiency
• boundary conditions

Innovation

• Variational Green's Function
• Neural Fields
• Boundary Conditions
• PDE Solvers
• Singularity Handling