# Learning two-phase microstructure evolution using neural operators and autoencoder architectures

**Authors:** Vivek Oommen, Khemraj Shukla, Somdatta Goswami, Remi Dingreville, George Em Karniadakis  
**arXiv:** [2204.07230](https://arxiv.org/abs/2204.07230)  
**Date:** 2022-04-11  
**Summary page:** https://www.summarizepaper.com/en/paper/2204.07230v2/

## Key points

- Phase-field modeling is a computationally expensive method for capturing microstructure evolution in materials
- Surrogate models can be used to accelerate prediction of microstructure evolution and reduce computational costs
- The proposed framework combines a convolutional autoencoder with a deep neural operator (DeepONet)
- The autoencoder represents microstructure data in a low-dimensional latent space
- DeepONet captures the mesoscale dynamics of microstructure evolution using the latent space generated by the autoencoder
- The trained DeepONet can replace high-fidelity phase-field numerical solvers in interpolation tasks and accelerate numerical solvers in extrapolation tasks
- The framework focuses on purely data-driven learning without explicit physics-based constraints, but these constraints could potentially be integrated for enhanced performance and consistency with physics principles
- This approach improves efficiency and reduces computational costs in materials science optimization and design processes.

## Summary

Phase-field modeling is a powerful but computationally expensive method for capturing the evolution of microstructures in materials. To address this issue, researchers have been exploring the use of surrogate models to accelerate the prediction of microstructure evolution and reduce computational costs. In this study, the authors propose a framework that combines a convolutional autoencoder architecture with a deep neural operator (DeepONet) to learn and predict the dynamic evolution of a two-phase mixture. The convolutional autoencoder is used to represent the microstructure data in a low-dimensional latent space, providing a compact representation of the complex morphological features. The DeepONet consists of two sub-networks: one for encoding the input function at specific sensor locations (branch net) and another for encoding the output function locations (trunk net). By learning from the latent space generated by the autoencoder, DeepONet captures the mesoscale dynamics of microstructure evolution. The decoder part of the convolutional autoencoder then reconstructs the time-evolved microstructure based on DeepONet predictions. The trained DeepONet architecture can be utilized in various ways. It can replace high-fidelity phase-field numerical solvers in interpolation tasks, where it accurately predicts missing data points based on known information. Additionally, it can also accelerate numerical solvers in extrapolation tasks, where it provides fast approximations for predicting future states beyond available data. It is worth noting that this framework focuses on purely data-driven learning without incorporating explicit physics-based constraints. However, similar to previous studies, physical constraints could potentially be integrated into this framework to enhance its performance and ensure consistency with underlying physics principles. Overall, this study presents an innovative approach to accelerate and predict microstructure evolution using a combination of convolutional autoencoders and deep neural operators. By leveraging these techniques, researchers can significantly reduce computational costs and improve the efficiency of optimization and design processes in materials science.