CloudNFMM: A Hybrid Hierarchical and Local Neural Operator Inspired by the Fast Multipole Method

ICLR 2026 Conference Submission19805 Authors

Keywords: Transformer, Neural Operators, Fast Multipole Method, Scientific Machine Learning, Partial Differential Equations, Hierarchical Methods, Point Cloud

TL;DR: A hybrid hierarchical and local Neural Operator designed after the information flow of the FMM.

Abstract:

The Fast Multipole Method (FMM) is an efficient numerical algorithm used to calculate long-range forces in many-body problems, leveraging hierarchical data structures and series expansions.
In this work, we present the Cloud Neural FMM (CloudNFMM), a new neural operator architecture that integrates the hierarchical structure of the FMM to learn the Green's operator of elliptic PDEs on point cloud data.
The architecture efficiently learns representations for both local and far-field interactions.
The core innovation is the local attention, a specialised local attention mechanism which models complex dependencies within a small neighbourhood of points.
We demonstrate the effectiveness of this approach, and discuss possible extensions and modifications to the CloudNFMM architecture.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 19805