# QuadConv: Quadrature-Based Convolutions with Applications to Non-Uniform PDE Data Compression

arXiv: https://arxiv.org/abs/2211.05151 · HTML: https://mlsanity.com/paper/2211.05151/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 51 references that are in the map.
- First posted on arXiv 9 Nov 2022.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [PointConv: Deep Convolutional Networks on 3D Point Clouds](https://mlsanity.com/paper/1811.07246/) (2018)
- [Deep Learning for 3D Point Clouds: A Survey](https://mlsanity.com/paper/1912.12033/) (2019)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Searching Efficient 3D Architectures with Sparse Point-Voxel Convolution](https://mlsanity.com/paper/2007.16100/) (2020)
- [Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs](https://mlsanity.com/paper/1704.02901/) (2017)
- [Generating 3D faces using Convolutional Mesh Autoencoders](https://mlsanity.com/paper/1807.10267/) (2018)
- [SpiderCNN: Deep Learning on Point Sets with Parameterized Convolutional Filters](https://mlsanity.com/paper/1803.11527/) (2018)
- [SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels](https://mlsanity.com/paper/1711.08920/) (2017)
- [Deep Parametric Continuous Convolutional Neural Networks](https://mlsanity.com/paper/2101.06742/) (2021)
- [CKConv: Continuous Kernel Convolution For Sequential Data](https://mlsanity.com/paper/2102.02611/) (2021)
- [MeshCNN: A Network with an Edge](https://mlsanity.com/paper/1809.05910/) (2018)
- [A-CNN: Annularly Convolutional Neural Networks on Point Clouds](https://mlsanity.com/paper/1904.08017/) (2019)
- [FlexConv: Continuous Kernel Convolutions with Differentiable Kernel Sizes](https://mlsanity.com/paper/2110.08059/) (2021)
- [ConvPoint: Continuous Convolutions for Point Cloud Processing](https://mlsanity.com/paper/1904.02375/) (2019)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [S4ND: Modeling Images and Videos as Multidimensional Signals Using State Spaces](https://mlsanity.com/paper/2210.06583/) (2022)
- [A Physics-Informed Vector Quantized Autoencoder for Data Compression of Turbulent Flow](https://mlsanity.com/paper/2201.03617/) (2022)
- [A Continuous Convolutional Trainable Filter for Modelling Unstructured Data](https://mlsanity.com/paper/2210.13416/) (2022)
- [Task-parallel in-situ temporal compression of large-scale computational fluid dynamics data](https://mlsanity.com/paper/2103.01380/) (2021)
- [Emulating Spatio-Temporal Realizations of Three-Dimensional Isotropic Turbulence via Deep Sequence Learning Models](https://mlsanity.com/paper/2112.03469/) (2021)

## Built on by (most cited first, 6 of 6)

- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Attention Based Machine Learning Methods for Data Reduction with Guaranteed Error Bounds](https://mlsanity.com/paper/2409.05357/) (2024)
- [In Situ Framework for Coupling Simulation and Machine Learning with Application to CFD](https://mlsanity.com/paper/2306.12900/) (2023)
- [Machine Learning Techniques for Data Reduction of CFD Applications](https://mlsanity.com/paper/2404.18063/) (2024)
- [GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains](https://mlsanity.com/paper/2605.03497/) (2026)
- [In Situ Training of Implicit Neural Compressors for Scientific Simulations via Sketch-Based Regularization](https://mlsanity.com/paper/2511.02659/) (2025)

## Abstract (arXiv)

We present a new convolution layer for deep learning architectures which we call QuadConv -- an approximation to continuous convolution via quadrature. Our operator is developed explicitly for use on non-uniform, mesh-based data, and accomplishes this by learning a continuous kernel that can be sampled at arbitrary locations. Moreover, the construction of our operator admits an efficient implementation which we detail and construct. As an experimental validation of our operator, we consider the task of compressing partial differential equation (PDE) simulation data from fixed meshes. We show that QuadConv can match the performance of standard discrete convolutions on uniform grid data by comparing a QuadConv autoencoder (QCAE) to a standard convolutional autoencoder (CAE). Further, we show that the QCAE can maintain this accuracy even on non-uniform data. In both cases, QuadConv also outperforms alternative unstructured convolution methods such as graph convolution.