# Accelerating phase-field-based simulation via machine learning

arXiv: https://arxiv.org/abs/2205.02121 · HTML: https://mlsanity.com/paper/2205.02121/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 51 references that are in the map.
- First posted on arXiv 4 May 2022.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Machine learning materials physics: Multi-resolution neural networks learn the free energy and nonlinear elastic response of evolving microstructures](https://mlsanity.com/paper/2001.01575/) (2019)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)

## Built on by (most cited first, 1 of 1)

- [Teaching Artificial Intelligence to Perform Rapid, Resolution-Invariant Grain Growth Modeling via Fourier Neural Operator](https://mlsanity.com/paper/2503.14568/) (2025)

## Abstract (arXiv)

Phase-field-based models have become common in material science, mechanics, physics, biology, chemistry, and engineering for the simulation of microstructure evolution. Yet, they suffer from the drawback of being computationally very costly when applied to large, complex systems. To reduce such computational costs, a Unet-based artificial neural network is developed as a surrogate model in the current work. Training input for this network is obtained from the results of the numerical solution of initial-boundary-value problems (IBVPs) based on the Fan-Chen model for grain microstructure evolution. In particular, about 250 different simulations with varying initial order parameters are carried out and 200 frames of the time evolution of the phase fields are stored for each simulation. The network is trained with 90% of this data, taking the i-th frame of a simulation, i.e. order parameter field, as input, and producing the (i+1)-th frame as the output. Evaluation of the network is carried out with a test dataset consisting of 2200 microstructures based on different configurations than originally used for training. The trained network is applied recursively on initial order parameters to calculate the time evolution of the phase fields. The results are compared to the ones obtained from the conventional numerical solution in terms of the errors in order parameters and the system's free energy. The resulting order parameter error averaged over all points and all simulation cases is 0.005 and the relative error in the total free energy in all simulation boxes does not exceed 1%.