Teaching Artificial Intelligence to Perform Rapid, Resolution-Invariant Grain Growth Modeling via Fourier Neural Operator

TL;DR

A novel approach utilizing Fourier Neural Operator (FNO) to achieve resolution-invariant modeling of microstructure evolution in multi-grain systems and demonstrates remarkable accuracy in predicting long-term evolution, even for unseen configurations and higher-resolution grids not encountered during training.

Abstract

Microstructural evolution, particularly grain growth, plays a critical role in shaping the physical, optical, and electronic properties of materials. Traditional phase-field modeling accurately simulates these phenomena but is computationally intensive, especially for large systems and fine spatial resolutions. While machine learning approaches have been employed to accelerate simulations, they often struggle with resolution dependence and generalization across different grain scales. This study introduces a novel approach utilizing Fourier Neural Operator (FNO) to achieve resolution-invariant modeling of microstructure evolution in multi-grain systems. FNO operates in the Fourier space and can inherently handle varying resolutions by learning mappings between function spaces. By integrating FNO with the phase field method, we developed a surrogate model that significantly reduces computational costs while maintaining high accuracy across different spatial scales. We generated a comprehensive dataset from phase-field simulations using the Fan Chen model, capturing grain evolution over time. Data preparation involved creating input-output pairs with a time shift, allowing the model to predict future microstructures based on current and past states. The FNO-based neural network was trained using sequences of microstructures and demonstrated remarkable accuracy in predicting long-term evolution, even for unseen configurations and higher-resolution grids not encountered during training.

Figures (11)

• Figure 1: Illustration of order parameters in a multi-grain system using the Fan–Chen phase-field model. Each grain is represented by a distinct non-conserved order parameter $\eta_i$ that equals 1 within its respective grain and 0 elsewhere.
• Figure 2: Example of grain evolution in a 2D phase-field simulation using the Fan–Chen model. Initial grain configurations are generated from random Voronoi diagrams and encoded using one-hot encoding. The figure displays frames at various time steps, illustrating how grains compete and evolve over time. Frame 72 shows the order parameter $\eta$ after 72,000 time steps, highlighting the microstructural changes during the simulation. Evolution of grain structure over time, illustrating the disappearance of an inner grain due to curvature-driven grain boundary migration
• Figure 3: Examples of input (x) and output (y) microstructures used for training the neural network. The input sequence consists of microstructures from time steps $t$ to $t+4$, and the output sequence consists of microstructures from time steps $t+10$ to $t+14$, demonstrating a time shift of $S = 10$ and a sequence length of $T = 5$. This configuration allows the model to learn to predict future microstructures based on current and past states, facilitating the capture of temporal dependencies in microstructural evolution.
• Figure 4: Impact of the number of retained Fourier modes on the model's performance and training time. The figure illustrates how varying the number of modes affects the average training loss over the last 10 epochs (blue line) and the total training time over 100 epochs (red line). As the number of modes increases, the training loss decreases due to the model's enhanced capacity to learn complex patterns, but the training time correspondingly increases because of the added computational burden. In our study, retaining 20 Fourier modes—with 2,564,917 learnable parameters—provided an optimal balance between accuracy and efficiency.
• Figure 5: Architecture of the FNO used in this study. The network begins with a fully connected layer that lifts the input data to a higher-dimensional feature space with 20 channels. This is followed by four consecutive spectral convolution layers, each interleaved with pointwise convolution layers. In the spectral convolution layers, the data is transformed into the Fourier domain using the FFT. Within this domain, the network selectively retains a specified number of lower-frequency Fourier modes that capture the most significant patterns in the data, while discarding higher-frequency modes that may contribute less to the overall structure or represent noise. After mode truncation, the data is transformed back into the spatial domain using the inverse FFT. This architecture enables efficient learning of both global and local features in the microstructural evolution