# GMLS-Nets: A framework for learning from unstructured data

arXiv: https://arxiv.org/abs/1909.05371 · HTML: https://mlsanity.com/paper/1909.05371/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2019.

## Built on by (most cited first, 18 of 18)

- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Mesh-based graph convolutional neural networks for modeling materials with microstructure](https://mlsanity.com/paper/2107.00090/) (2021)
- [Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Data](https://mlsanity.com/paper/2303.11379/) (2023)
- [Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions](https://mlsanity.com/paper/2307.02478/) (2023)
- [3D Neural Operator-Based Flow Surrogates around 3D geometries: Signed Distance Functions and Derivative Constraints](https://mlsanity.com/paper/2503.17289/) (2025)
- [Discretization-independent surrogate modeling over complex geometries using hypernetworks and implicit representations](https://mlsanity.com/paper/2109.07018/) (2021)
- [Error-in-variables modelling for operator learning](https://mlsanity.com/paper/2204.10909/) (2022)
- [Stratospheric aerosol source inversion: Noise, variability, and uncertainty quantification](https://mlsanity.com/paper/2409.06846/) (2024)
- [Neural Network with Local Converging Input (NNLCI) for Supersonic Flow Problems with Unstructured Grids](https://mlsanity.com/paper/2310.15299/) (2023)
- [A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds](https://mlsanity.com/paper/2605.08436/) (2026)

## Abstract (arXiv)

Data fields sampled on irregularly spaced points arise in many applications in the sciences and engineering. For regular grids, Convolutional Neural Networks (CNNs) have been successfully used to gaining benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds based on Generalized Moving Least Squares (GMLS). GMLS is a non-parametric technique for estimating linear bounded functionals from scattered data, and has recently been used in the literature for solving partial differential equations. By parameterizing the GMLS estimator, we obtain learning methods for operators with unstructured stencils. In GMLS-Nets the necessary calculations are local, readily parallelizable, and the estimator is supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform functional regression to identify associated differential operators and to regress quantities of interest. The results suggest the architectures to be an attractive foundation for data-driven model development in scientific machine learning applications.