Neural Green's Functions¶

Conference: NeurIPS 2025 arXiv: 2511.01924 Code: None Area: 3D Vision Keywords: Green's functions, neural operators, PDE solving, eigendecomposition, domain generalization

TL;DR¶

This paper proposes Neural Green's Functions, a learnable linear PDE solution operator based on eigendecomposition: pointwise geometric features are extracted from the domain geometry to predict the eigendecomposition of the Green's function, enabling one-time training to solve for arbitrary source functions and boundary conditions via numerical integration. On mechanical part thermal analysis, the method reduces error by 13.9% over the state-of-the-art neural operator while running 350× faster than numerical solvers.

Background & Motivation¶

Background: Learning-based PDE solvers (PINNs, FNO, GNO, Transolver, etc.) have significantly improved efficiency, yet face fundamental difficulties in simultaneously generalizing across varying domain geometries, source functions, and boundary conditions.

Limitations of Prior Work: - PINNs require independent training for each problem instance; any change necessitates retraining. - Neural operators (FNO, Transolver) couple the input mesh with sampled function values — generalization degrades when the function changes. - Prior methods for learning Green's functions (Boullé et al., Teng et al.) are restricted to a single domain and handle only simple geometries.

Key Challenge: Existing methods take function values as network inputs — changing the function demands more training samples.

Key Insight: The Green's function $G_D(x,y)$ depends only on the domain geometry $D$, independent of the source function $f$ and boundary condition $h$. Once $G_D$ is learned, the solution is obtained by integration: $u(x) = \int G_D(x,y) f(y)\, dy + \text{boundary terms}$.

Core Idea: A neural network operating on point clouds predicts the eigendecomposition of the Green's function $G \approx \Phi \Lambda^{-1} \Phi^T$ from domain geometry alone, along with the mass matrix $M$ required for integration — making the design function-agnostic by construction.

Method¶

Overall Architecture¶

Input: Volumetric point cloud representation of the problem domain $D$. The network extracts pointwise geometric features → predicts eigenvectors $\Phi$ and eigenvalues $\Lambda$ of the Green's function, along with the mass matrix $M$ and the boundary-interior selection matrix. For any given $f$ and $h$, the solution is obtained directly via matrix operations: $$\mathbf{u} = \mathbf{K}^T \{\mathbf{G}(\mathbf{K}\mathbf{M}\mathbf{f} - \mathbf{K}\mathbf{L}\mathbf{S}^T\mathbf{h})\} + \mathbf{S}^T\mathbf{h}$$

Key Designs¶

Eigendecomposition-Based Solution Operator:
Function: Factorizes the Green's function matrix $\mathbf{G} = (\mathbf{KLK}^T)^{-1}$ as $\mathbf{G} = \mathbf{\Phi} \mathbf{\Lambda}^{-1} \mathbf{\Phi}^T$.
Mechanism: The network predicts eigenvectors $\Phi \in \mathbb{R}^{N_{int} \times K}$ and eigenvalues $\Lambda$, where $K$ is far smaller than the number of interior vertices $N_{int}$ — a low-rank approximation.
Design Motivation: Directly predicting the $N_{int} \times N_{int}$ Green's function matrix is infeasible (too large); the low-rank eigendecomposition compresses parameters from $O(N^2)$ to $O(NK)$.
Purely Geometric Feature Extraction:
Function: Extracts pointwise features from the input point cloud without using any function value information.
Mechanism: A Transolver backbone processes the volumetric point cloud, with cross-attention interactions with latent tokens.
Design Motivation: Forcing the network to observe only geometry is the key design choice that aligns with the mathematical property that Green's functions depend solely on domain geometry.
Joint Prediction of Integration Components:
Mass matrix $M$: Converts source function values into integration weights for irregular meshes.
Selection matrices $S$/$K$: Distinguish boundary from interior vertices.
All components are predicted from geometry — mesh density and shape vary across domains, so $M$ and $S$ vary accordingly.

Loss & Training¶

Supervised loss: MSE between the predicted solution $u$ and the ground truth from a numerical solver.
Training uses the MCB dataset of mechanical part 3D geometries across 5 categories.
Linear PDEs considered: Poisson equation and Biharmonic equation.

Key Experimental Results¶

Main Results — Steady-State Thermal Analysis (MCB Dataset, 5 Part Categories)¶

Method	Mean Relative Error ↓	Generalizes to New Functions	Generalizes to New Domains	Speed
PINN	Requires retraining	✗	✗	Slow
FNO	Moderate	Poor	Poor	Fast
GINO	Moderate	Poor	Moderate	Fast
Transolver	Baseline	Poor	Moderate	Fast
Neural Green's	−13.9%	✓ (by design)	✓	Fast (350× vs. FEM)

Ablation Study¶

Configuration	Error	Notes
Full framework	Lowest	Eigendecomposition + mass matrix
Replace with direct regression of $u$	+15%	Loses function-agnosticism
Remove mass matrix prediction	+8%	Inaccurate integration weights
Reduce eigenvector dimension $K$	+5–20%	Information loss when $K$ is too small

Validation on Poisson/Biharmonic Equations in Simple Domains¶

PDE	Method	Error on New Source Functions	Error on New Boundary Conditions
Poisson	FNO	Moderate	Moderate
Poisson	Neural Green's	Low	Low
Biharmonic	Transolver	Moderate	Moderate
Biharmonic	Neural Green's	Low	Low

Key Findings¶

Function-agnosticism is thoroughly validated experimentally: performance degradation on source functions and boundary conditions unseen during training is minimal (<5%), whereas baseline methods degrade by 20–50%.
The low-rank approximation of the eigendecomposition ($K = 50$–$100$) is sufficient to capture the majority of information.
The method is 350× faster than numerical solvers (which require meshing and linear system solving), eliminating the meshing bottleneck.
Cross-category generalization (training on one shape category, testing on another) also holds, indicating that the network learns a general geometry-to-solution-operator mapping.

Highlights & Insights¶

"Learning the fundamental solution rather than a specific solution" reflects a profound problem decomposition — the Green's function is the inverse operator of the PDE; knowing it yields all solutions. This is fundamentally more powerful than learning a solution for a particular $f$/$h$.
Design aligned with mathematical structure: Green's functions depend only on geometry → the network observes only geometry. The linear structure of eigendecomposition → the network predicts eigenvectors and eigenvalues. This methodology of letting mathematical properties guide network design generalizes naturally to other operator learning problems.
The effectiveness of the low-rank approximation suggests that the Green's functions of most PDEs are intrinsically low-rank — consistent with physical intuition that the influence of distant sources decays rapidly.
A 13.9% error reduction over Transolver (same backbone) is attributable purely to framework design.

Limitations & Future Work¶

Applicable only to linear PDEs whose dense operators admit eigendecomposition (e.g., Poisson, Biharmonic); nonlinear PDEs (e.g., Navier–Stokes) are not directly supported.
The current formulation handles Dirichlet boundary conditions; Neumann or mixed boundary conditions require modifications to the boundary terms of the Green's function.
Point cloud representations of 3D volumetric meshes and mass matrix prediction may suffer accuracy issues at very high resolutions.
Only steady-state PDEs are validated; time-dependent PDEs (e.g., the heat equation) require extension to the temporal-domain version of the Green's function.

vs. Boullé et al. (2021): Their approach learns the Green's function for a fixed domain (requiring retraining per domain), whereas this work learns a geometry-to-Green's-function mapping across domains.
vs. Transolver (ICML'24): Same backbone but different framework — Transolver takes function values as input, while this work observes only geometry. The performance gap stems entirely from framework design.
vs. DeepONet: DeepONet learns general operators but encodes source functions through a branch network — generalizing to new functions requires sufficient training samples.
Inspiration: For other physical problems (e.g., electromagnetics, elasticity), similar eigendecomposition-based methods may enable function-agnostic solution operators.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ A triple innovation combining Green's functions, eigendecomposition, and geometric priors; the first Green's function learning framework that generalizes across both domains and functions.
Experimental Thoroughness: ⭐⭐⭐⭐ Validated on both simple domains and complex 3D mechanical parts, with comparisons against multiple state-of-the-art methods.
Writing Quality: ⭐⭐⭐⭐⭐ Rigorous mathematical derivations with clear exposition of the alignment between network design and physical/mathematical structure.
Value: ⭐⭐⭐⭐⭐ Transformative potential for AI-driven scientific computing — train once, solve for anything.