# Learning Greens Operators through Hierarchical Neural Networks Inspired by the Fast Multipole Method

arXiv: https://arxiv.org/abs/2509.20591 · HTML: https://mlsanity.com/paper/2509.20591/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 24 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [BCR-Net: a neural network based on the nonstandard wavelet form](https://mlsanity.com/paper/1810.08754/) (2018)
- [Solving Inverse Wave Scattering with Deep Learning](https://mlsanity.com/paper/1911.13202/) (2019)
- [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023)
- [A Learned Born Series for Highly-Scattering Media](https://mlsanity.com/paper/2212.04948/) (2022)

## Built on by (most cited first, 2 of 2)

- [Structure-Adaptive Tree Field Integrators](https://mlsanity.com/paper/2609.34025/) (2026)
- [NEMORA: Neural Equivariant Multipole Operators for Long-Range Atomistic Learning](https://mlsanity.com/paper/2610.10776/) (2026)

## Abstract (arXiv)

The Fast Multipole Method (FMM) is an efficient numerical algorithm for computation of long-ranged forces in N-body problems within gravitational and electrostatic fields. This method utilizes multipole expansions of the Green's function inherent to the underlying dynamical systems. Despite its widespread application in physics and engineering, the integration of FMM with modern machine learning architectures remains underexplored. In this work, we propose a novel neural network architecture, the Neural FMM, that integrates the information flow of the FMM into a hierarchical machine learning framework for learning the Green's operator of an Elliptic PDE. Our Neural FMM architecture leverages a hierarchical computation flow of the FMM method to split up the local and far-field interactions and efficiently learn their respective representations.