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

2025-09-14

AI Summary

Addressing the challenges of solving and classifying multiphase solutions in the Lifshitz–Petrich (LP) model—arising from high-order gradient terms and quasicrystalline long-range orientational order—this work proposes a derivative-enhanced graph convolutional autoencoder framework. In the offline phase, the method jointly encodes solution fields and their spatial derivatives to construct graph-structured data, leveraging graph convolutions to capture nonlocal spatial dependencies and an autoencoder for effective dimensionality reduction of high-dimensional solution manifolds. In the online phase, a lightweight feedforward network enables rapid phase classification. Experiments demonstrate that the approach significantly improves robustness and accuracy in identifying multicomponent, multimorphological solutions, efficiently generating high-resolution phase diagrams. It outperforms conventional numerical solvers and standard machine learning methods in both computational efficiency and classification accuracy.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

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.

Problem

• Classifying multi-component multi-state solutions of Lifshitz-Petrich model
• Addressing high-order gradient terms and long-range orientational order
• Reducing dimensionality while capturing intricate spatial dependencies

Innovation

• Derivative-informed Graph Convolutional Autoencoder for solution classification
• Incorporates solutions and derivatives to capture spatial dependencies
• Two-stage framework with autoencoder and neural network classifier