# Learning Singularity-Encoded Green's Functions with Application to Iterative Methods

arXiv: https://arxiv.org/abs/2509.11580 · HTML: https://mlsanity.com/paper/2509.11580/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 60 references that are in the map.
- First posted on arXiv 15 Sep 2025.

## Builds on (most cited first, 26 of 26)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2012.08023/) (2020)
- [A cusp-capturing PINN for elliptic interface problems](https://mlsanity.com/paper/2210.08424/) (2022)
- [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021)
- [A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples](https://mlsanity.com/paper/2402.07156/) (2024)
- [Neural Green's Operators for Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.01857/) (2024)
- [Neural Preconditioning Operator for Efficient PDE Solves](https://mlsanity.com/paper/2502.01337/) (2025)
- [Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method](https://mlsanity.com/paper/2209.02931/) (2022)
- [Green Multigrid Network](https://mlsanity.com/paper/2407.03593/) (2024)
- [Learning Green's Function Efficiently Using Low-Rank Approximations](https://mlsanity.com/paper/2308.00350/) (2023)
- [A categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems](https://mlsanity.com/paper/2503.15441/) (2025)

## Built on by (most cited first, 3 of 3)

- [Are Deep Learning Based Hybrid PDE Solvers Reliable? Why Training Paradigms and Update Strategies Matter](https://mlsanity.com/paper/2602.06842/) (2026)
- [The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions](https://mlsanity.com/paper/2602.21910/) (2026)
- [SDC-GON: Singular Decomposition and Consistency-Regularized Green's Operator Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2609.23206/) (2026)

## Abstract (arXiv)

Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the design of effective solvers. Unfortunately, numerical computation of Green's function remains challenging due to its doubled dimensionality and intrinsic singularity. In this paper, we present a novel singularity-encoded learning approach to resolve these problems in an unsupervised fashion. Our method embeds the Green's function within a one-order higher-dimensional space by encoding its prior estimate as an augmented variable, followed by a neural network parametrization to manage the increased dimensionality. By projecting the trained neural network solution back onto the original domain, our deep surrogate model exploits its spectral bias to accelerate conventional iterative schemes, serving either as a preconditioner or as part of a hybrid solver. The effectiveness of our proposed method is empirically verified through numerical experiments with two and four dimensional Green's functions, achieving satisfactory resolution of singularities and acceleration of iterative solvers.