# A Tailored Convolutional Neural Network for Nonlinear Manifold Learning of Computational Physics Data using Unstructured Spatial Discretizations

arXiv: https://arxiv.org/abs/2006.06154 · HTML: https://mlsanity.com/paper/2006.06154/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2020.

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

- [Neural Physics: Using AI Libraries to Develop Physics-Based Solvers for Incompressible Computational Fluid Dynamics](https://mlsanity.com/paper/2402.17913/) (2024)
- [An AI-based Domain-Decomposition Non-Intrusive Reduced-Order Model for Extended Domains applied to Multiphase Flow in Pipes](https://mlsanity.com/paper/2202.06170/) (2022)
- [M2M: Learning controllable Multi of experts and multi-scale operators are the Partial Differential Equations need](https://mlsanity.com/paper/2410.11617/) (2024)
- [Systematic construction of continuous-time neural networks for linear dynamical systems](https://mlsanity.com/paper/2403.16215/) (2024)
- [Hyper-Reduced Autoencoders for Efficient and Accurate Nonlinear Model Reductions](https://mlsanity.com/paper/2303.09630/) (2023)
- [Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks](https://mlsanity.com/paper/2605.27756/) (2026)

## Abstract (arXiv)

We propose a nonlinear manifold learning technique based on deep convolutional autoencoders that is appropriate for model order reduction of physical systems in complex geometries. Convolutional neural networks have proven to be highly advantageous for compressing data arising from systems demonstrating a slow-decaying Kolmogorov n-width. However, these networks are restricted to data on structured meshes. Unstructured meshes are often required for performing analyses of real systems with complex geometry. Our custom graph convolution operators based on the available differential operators for a given spatial discretization effectively extend the application space of deep convolutional autoencoders to systems with arbitrarily complex geometry that are typically discretized using unstructured meshes. We propose sets of convolution operators based on the spatial derivative operators for the underlying spatial discretization, making the method particularly well suited to data arising from the solution of partial differential equations. We demonstrate the method using examples from heat transfer and fluid mechanics and show better than an order of magnitude improvement in accuracy over linear methods.