# NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data

arXiv: https://arxiv.org/abs/2305.18694 · HTML: https://mlsanity.com/paper/2305.18694/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 39 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition](https://mlsanity.com/paper/1406.4729/) (2014)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications](https://mlsanity.com/paper/2004.04276/) (2020)

## Built on by (most cited first, 15 of 15)

- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2510.25803/) (2025)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Towards General Neural Surrogate Solvers with Specialized Neural Accelerators](https://mlsanity.com/paper/2405.02351/) (2024)
- [Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving](https://mlsanity.com/paper/2504.00510/) (2025)
- [Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations](https://mlsanity.com/paper/2405.17509/) (2024)
- [Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models](https://mlsanity.com/paper/2505.10930/) (2025)
- [GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2508.04463/) (2025)
- [Enhancing Solutions for Complex PDEs: Introducing Complementary Convolution and Equivariant Attention in Fourier Neural Operators](https://mlsanity.com/paper/2311.12902/) (2023)
- [Benchmarking Long Roll-outs of Auto-regressive Neural Operators for the Compressible Navier-Stokes Equations with Conserved Quantity Correction](https://mlsanity.com/paper/2601.22541/) (2026)
- [Stabilizing autoregressive forecasts in chaotic systems via multi-rate latent recurrence](https://mlsanity.com/paper/2601.14487/) (2026)
- [UniFluids: Unified Neural Operator Learning with Conditional Flow-matching](https://mlsanity.com/paper/2603.22309/) (2026)
- [Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2603.24143/) (2026)
- [Geometry-aware Latent Autoregressive Generative Model for PDEs in Complex Domains](https://mlsanity.com/paper/2609.00297/) (2026)
- [Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators](https://mlsanity.com/paper/2609.05860/) (2026)

## Abstract (arXiv)

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.