# Applying Convolutional Neural Networks to Data on Unstructured Meshes with Space-Filling Curves

arXiv: https://arxiv.org/abs/2011.14820 · HTML: https://mlsanity.com/paper/2011.14820/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

## Built on by (most cited first, 8 of 8)

- [Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models](https://mlsanity.com/paper/2204.03497/) (2022)
- [Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques](https://mlsanity.com/paper/2107.11460/) (2021)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Adaptive learning of effective dynamics: Adaptive real-time, online modeling for complex systems](https://mlsanity.com/paper/2304.01732/) (2023)
- [Neural Physics: Using AI Libraries to Develop Physics-Based Solvers for Incompressible Computational Fluid Dynamics](https://mlsanity.com/paper/2402.17913/) (2024)
- [Out-of-distribution generalization of deep-learning surrogates for 2D PDE-generated dynamics in the small-data regime](https://mlsanity.com/paper/2601.08404/) (2026)
- [Twisted Convolutional Networks (TCNs): Enhancing Feature Interactions for Non-Spatial Data Classification](https://mlsanity.com/paper/2412.00238/) (2024)
- [Learning Mappings in Mesh-based Simulations](https://mlsanity.com/paper/2506.12652/) (2025)

## Abstract (arXiv)

This paper presents the first classical Convolutional Neural Network (CNN) that can be applied directly to data from unstructured finite element meshes or control volume grids. CNNs have been hugely influential in the areas of image classification and image compression, both of which typically deal with data on structured grids. Unstructured meshes are frequently used to solve partial differential equations and are particularly suitable for problems that require the mesh to conform to complex geometries or for problems that require variable mesh resolution. Central to the approach are space-filling curves, which traverse the nodes or cells of a mesh tracing out a path that is as short as possible (in terms of numbers of edges) and that visits each node or cell exactly once. The space-filling curves (SFCs) are used to find an ordering of the nodes or cells that can transform multi-dimensional solutions on unstructured meshes into a one-dimensional (1D) representation, to which 1D convolutional layers can then be applied. Although developed in two dimensions, the approach is applicable to higher dimensional problems. To demonstrate the approach, the network we choose is a convolutional autoencoder (CAE) although other types of CNN could be used. The approach is tested by applying CAEs to data sets that have been reordered with an SFC. Sparse layers are used at the input and output of the autoencoder, and the use of multiple SFCs is explored. We compare the accuracy of the SFC-based CAE with that of a classical CAE applied to two idealised problems on structured meshes, and then apply the approach to solutions of flow past a cylinder obtained using the finite-element method and an unstructured mesh.