# Neural Green's Functions

arXiv: https://arxiv.org/abs/2511.01924 · HTML: https://mlsanity.com/paper/2511.01924/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 37 references that are in the map.
- First posted on arXiv 2 Nov 2025.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Fast Tetrahedral Meshing in the Wild](https://mlsanity.com/paper/1908.03581/) (2019)
- [Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operators](https://mlsanity.com/paper/2402.12365/) (2024)
- [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021)
- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Learning Domain-Independent Green's Function For Elliptic Partial Differential Equations](https://mlsanity.com/paper/2401.17172/) (2024)

## Built on by (most cited first, 6 of 6)

- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems](https://mlsanity.com/paper/2607.01128/) (2026)
- [Von Neumann Networks](https://mlsanity.com/paper/2605.05780/) (2026)
- [Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields](https://mlsanity.com/paper/2605.28551/) (2026)
- [Neural Harmonic Measure Operator](https://mlsanity.com/paper/2609.35752/) (2026)
- [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026)

## Abstract (arXiv)

We introduce Neural Green's Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions. Inspired by Green's functions, the solution operators of linear PDEs that depend exclusively on the domain geometry, we design Neural Green's Function to imitate their behavior, achieving superior generalization across diverse irregular geometries and source and boundary functions. Specifically, Neural Green's Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict a decomposition of the solution operator, which is subsequently applied to evaluate solutions via numerical integration. Unlike recent learning-based solution operators, which often struggle to generalize to unseen source or boundary functions, our framework is, by design, agnostic to the specific functions used during training, enabling robust and efficient generalization. In the steady-state thermal analysis of mechanical part geometries from the MCB dataset, Neural Green's Function outperforms state-of-the-art neural operators, achieving an average error reduction of 13.9% across five shape categories, while being up to 350 times faster than a numerical solver that requires computationally expensive meshing.