# Derivative-informed Graph Convolutional Autoencoder with Phase Classification for the Lifshitz-Petrich Model

arXiv: https://arxiv.org/abs/2509.11293 · HTML: https://mlsanity.com/paper/2509.11293/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 35 references that are in the map.
- First posted on arXiv 14 Sep 2025.

## Builds on (most cited first, 11 of 11)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Variational Autoencoder for Deep Learning of Images, Labels and Captions](https://mlsanity.com/paper/1609.08976/) (2016)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Fully Convolutional Mesh Autoencoder using Efficient Spatially Varying Kernels](https://mlsanity.com/paper/2006.04325/) (2020)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [SpecAE: Spectral AutoEncoder for Anomaly Detection in Attributed Networks](https://mlsanity.com/paper/1908.03849/) (2019)
- [Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks](https://mlsanity.com/paper/2308.01602/) (2023)

## Abstract (arXiv)

The Lifshitz-Petrich (LP) model is a classical model for describing complex spatial patterns such as quasicrystals and multiphase structures. Solving and classifying the solutions of the LP model is challenging due to the presence of high-order gradient terms and the long-range orientational order characteristic of the quasicrystals. To address these challenges, we propose a Derivative-informed Graph Convolutional Autoencoder (DiGCA) to classify the multi-component multi-state solutions of the LP model. The classifier consists of two stages. In the offline stage, the DiGCA phase classifier innovatively incorporates both solutions and their derivatives for training a graph convolutional autoencoder which effectively captures intricate spatial dependencies while significantly reducing the dimensionality of the solution space. In the online phase, the framework employs a neural network classifier to efficiently categorize encoded solutions into distinct phase diagrams. The numerical results demonstrate that the DiGCA phase classifier accurately solves the LP model, classifies its solutions, and rapidly generates detailed phase diagrams in a robust manner, offering significant improvements in both efficiency and accuracy over traditional methods.