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[2602.12349] Variational Green's Functions for Volumetric PDEs
Summary
This article presents a novel method called Variational Green's Function (VGF) for efficiently computing Green's functions for volumetric partial differential equations (PDEs), enhancing applications in graphics and physical simulations.
Why It Matters
The development of Variational Green's Functions addresses the computational challenges associated with evaluating Green's functions on complex geometries. This advancement is significant for fields such as computer graphics and physical simulations, where accurate modeling of PDEs is crucial. By improving efficiency and differentiability, VGF can facilitate more complex simulations and analyses in various scientific and engineering applications.
Key Takeaways
- VGF provides a smooth, differentiable representation of Green's functions for linear self-adjoint PDE operators.
- The method effectively resolves sharp singularities by decomposing the Green's function into analytic and learned components.
- VGF naturally imposes Neumann boundary conditions and utilizes a projective layer for Dirichlet conditions.
- The resulting Green's functions are fast to evaluate and can be conditioned on other signals, enhancing their applicability.
- This approach has potential implications for improving simulations in graphics and other fields reliant on PDEs.
Computer Science > Graphics arXiv:2602.12349 (cs) [Submitted on 12 Feb 2026] Title:Variational Green's Functions for Volumetric PDEs Authors:Joao Teixeira, Eitan Grinspun, Otman Benchekroun View a PDF of the paper titled Variational Green's Functions for Volumetric PDEs, by Joao Teixeira and 2 other authors View PDF HTML (experimental) 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. Subjects: Graphics (cs.GR); Machine Learning (cs.LG) Cite as: arXiv:2602....