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Graph Neural Networks for an Accurate and Interpretable Prediction of the Properties of Polycrystalline Materials
Minyi Dai, Mehmet F. Demirel, Yingyu Liang, Jia-Mian Hu
TL;DR
Existing polycrystalline-material models do not directly account for physical interactions among neighboring grains, despite their importance to macroscopic properties. The paper represents microstructures as graphs, uses a graph neural network to embed grain features and interactions, and links that embedding to magnetostriction prediction. The model achieves low relative error across diverse Terfenol-D microstructures while quantifying the importance of individual grain features.
Problem
Existing models do not directly consider neighboring-grain physical interactions, although such microscopic interactions critically determine polycrystalline macroscopic properties.
Method
A graph neural network embeds polycrystalline microstructures using grain features and their interactions, then a fully connected network links the embedding to the target property.
Results
8.05% relative error is achieved for property prediction, with approximately 10% error maintained across a broad dataset-size range.
Takeaways & Limitations
The microstructure-graph GNN enables accurate and interpretable prediction of polycrystalline properties by quantifying the importance of each grain feature.
Takeaways & Limitations
The appropriate update functions depend critically on the physical principles of the target material system or properties, making one fixed model challenging to determine.
Abstract
from arXiv · showhide
Various machine learning models have been used to predict the properties of polycrystalline materials, but none of them directly consider the physical interactions among neighboring grains despite such microscopic interactions critically determining macroscopic material properties. Here, we develop a graph neural network (GNN) model for obtaining an embedding of polycrystalline microstructure which incorporates not only the physical features of individual grains but also their interactions. The embedding is then linked to the target property using a feed-forward neural network. Using the magnetostriction of polycrystalline Tb0.3Dy0.7Fe2 alloys as an example, we show that a single GNN model with fixed network architecture and hyperparameters allows for a low prediction error of ~10% over a group of remarkably different microstructures as well as quantifying the importance of each feature in each grain of a microstructure to its magnetostriction. Such microstructure-graph-based GNN model therefore enables an accurate and interpretable prediction of the properties of polycrystalline materials.
Introduction
Existing approaches often represent microstructures statistically or as images, but they do not directly preserve neighboring-grain adjacency and interactions. The paper introduces a microstructure-graph GNN that combines grain features with their interactions for property prediction and interpretation.
- Research gap: Neighboring-grain interactions are omitted by existing statistical-descriptor and image-based models even though they critically determine polycrystalline macroscopic properties.These models do not store grain adjacency, which limits direct treatment of microscopic interactions.
- Microstructure graph: The proposed model represents each grain as a graph node and stores grain features and physical-contact relations in a feature matrix F and adjacency matrix A.The graph input is G=(F, A), with adjacency indicating whether two grains are neighbors.
- Proposed approach: A graph neural network embeds polycrystalline microstructures by correlating physical features from neighboring grains before property regression.The approach links the graph embedding to the target property through a subsequent prediction model.
- Demonstration: Using Terfenol-D magnetostriction, the model achieves approximately 10% prediction error across different microstructures and quantifies each grain feature’s importance.The study uses polycrystalline Tb0.3Dy0.7Fe2 alloys as its example system.
Dataset Generation
The study generates diverse three-dimensional Terfenol-D microstructures and computes their field-dependent effective magnetostriction through phase-field modeling. The resulting graph-property data span broad grain counts, textures, magnetic fields, and local interaction effects.
- Microstructure generation: 492 generated 3D microstructures contain 12–297 grains each and include samples with and without strong textures.Their grain statistics cover size, neighborhood count, and orientation across 87,981 grains.
- Property calculation: Phase-field modeling computes local magnetization and magnetostriction distributions for each microstructure under applied magnetic fields.Effective magnetostriction is obtained by volumetrically averaging local magnetostriction.
- Field response: Effective magnetostriction increases with field as grain magnetization rotates toward the applied-field direction, then reaches saturation when the magnetizations align.The reference value is zero effective magnetostriction at zero field.
- Dataset: Four or five field values per microstructure produce 2,287 graph–field–magnetostriction data points for learning.Each data point has the form [(G, Hx), λ_xx^eff].
Graph Neural Network built upon Microstructure Graph
The model uses graph convolutional message passing to incorporate neighboring-grain features into a microstructure embedding, then regresses magnetostriction from that embedding and applied field. On the Terfenol-D dataset, prediction error is low even with relatively small training sets.
- Model rationale: The GCN embedding incorporates both individual grain features and their interactions, addressing a gap in prior polycrystalline-material applications.The paper uses a relatively simple graph convolutional network for this purpose.
- Message passing: Graph convolution updates each node from its own features and neighboring-node features while keeping the adjacency matrix fixed.The normalized adjacency operation produces weighted averages of node and neighboring feature vectors.
- Regression: The resulting embedding combines with the applied field as input to a fully connected layer that regresses the target property.Trainable weights and biases map the embedding and field to the predicted output.
- Dataset-size analysis: MARE decreases from 17% to approximately 10% with 72 microstructure graphs and then saturates near 9% as dataset size increases.The standard deviation also decreases and then stabilizes with larger datasets.
Computational efficiency of the GNN model
The GNN is computationally efficient because it processes grains as graph nodes rather than voxels. Compared with CNNs and phase-field simulations, it supports larger microstructures and rapid property prediction under the reported tests.
- Prediction speed: A trained GNN predicts 228 data points in approximately 0.2 seconds, whereas one phase-field data point takes about five hours on 16 cores.The comparison concerns property prediction after model training versus direct phase-field computation.
- Scaling behavior: The GNN training time does not vary with voxel count when the number of grains remains fixed, whereas CNN training time rises with voxel count.GNN representation is node-by-node, while CNN representation is voxel-by-voxel.
- CNN limitation: CNN cache memory becomes insufficient beyond 112^3, 120^3, and 64^3 voxels for CNN-1, CNN-2, and CNN-3, respectively.This constrains applying those 3D CNNs to very large microstructure image datasets.
- GNN versus CNN: A CNN-2 model takes about 35 times longer than the GNN for a 120^3-voxel microstructure containing 300 grains.The authors expect the training-time difference to increase for larger microstructure images.
- Large-scale microstructures: The GNN handles microstructures with up to 4,700 grains without exceeding one Tesla P100 GPU’s cache-memory limit.The reported total training time is approximately 13 hours for a 160-point training dataset.
Discussion
The discussion presents a GNN that represents polycrystalline microstructures as graphs, predicts magnetostriction accurately across diverse structures, and supports grain-level interpretation. It also highlights computational efficiency, extensibility, and material-dependent design choices and limitations.
- Discussion: 8.05% relative error was achieved for roughly 500 microstructures containing 12–297 grains, while prediction error remained near 10% across dataset sizes.A single fixed-architecture, fixed-hyperparameter GNN was used for these diverse microstructures.
- Discussion: Integrated Gradients quantified the importance of each physical feature in each grain, including grain-size contributions through a physically defined baseline graph.The baseline sets grain size to zero while preserving other node features, isolating its contribution to the macroscopic property.
- Discussion: Graph-based grain-level analysis is computationally more efficient than voxel-based CNN analysis, especially for large three-dimensional microstructures with billions of voxels.The efficiency advantage arises because one grain can occupy multiple voxels in a high-resolution image.
- Discussion: The framework can incorporate grain-boundary or defect features through edge vectors, but graph complexity, physical features, architecture, and update functions remain material-dependent.The paper states that no single GNN configuration is universally applicable across all polycrystalline materials and effective properties.
Phase-field simulations of the effective magnetostriction of polycrystals
Phase-field simulations model magnetization and elastic responses in discretized polycrystalline microstructures to obtain effective magnetostriction.
- Simulation setup: The simulations discretize each polycrystal into 64 × 64 × 64 cuboid cells and evolve local magnetization under applied magnetic fields.The magnetization dynamics use the Landau–Lifshitz–Gilbert equation with Fourier-spectral numerical methods.
- Energy model: The total free-energy density combines exchange, stray-field, external-field, anisotropy, and elastic contributions.The stray field uses periodic boundary conditions and depends on the demagnetizing factor and average magnetization.
- Crystalline anisotropy and strain: Crystal orientation is incorporated by transforming magnetization and spontaneous strain between local crystalline and global coordinate systems.Euler angles determine the rotation matrix relating the two systems.
- Elastic response: The total strain contains homogeneous and heterogeneous components, with the latter obtained by solving mechanical equilibrium numerically.With no external strain, homogeneous strain is determined by minimizing total elastic energy.
- Magnetostriction output: For a stress-free polycrystal, local magnetostriction equals the local spontaneous strain component under the simulation conditions.The effective magnetostriction is therefore derived from the equilibrated local magnetization and strain fields.
GNN model training and evaluation protocols
The GNN is trained and evaluated through gradient-based optimization, hyperparameter selection, held-out testing, cross-validation, and stability checks.
- Optimization: Hyperparameter optimization varies graph-layer dimensions, fully connected-layer widths, batch size, epochs, and learning rate.All trainable weights are updated through gradient descent.
- Training objective and metrics: The loss is mean square error, while macro average relative error measures relative prediction error on testing data.MARE compares predicted and simulated values over the testing points.
- Model selection: 81 GNN configurations are trained, and the model with the lowest validation MARE is applied to an unseen testing dataset.The data are divided into eight training, one validation, and one testing subset.
- Data ablation: Data ablation uses 10-fold cross-validation on randomly selected groups of 32 to 392 microstructure graphs.Each group is repeatedly split into training and testing subsets, and averaged MSE and MARE are reported.
- Stability analysis: Model stability is assessed across three random partitionings, producing average and standard-deviation MARE values from 30 testing errors.The analysis varies both microstructure selection and data-subset assignment.
Time measurement of the GNN models and CNN models
Training-time comparisons use matched CNN and GNN settings on 3D microstructure images and a common GPU environment.
- Controlled timing protocol: CNN and GNN timing comparisons use the same batch size and number of epochs, with data loading and training performed on one Tesla P100 GPU core.The three CNN models share the neural-network settings used for the comparison.
- CNN comparison: CNN inputs are Nd × Nd × Nd voxel images, with each voxel assigned three Euler-angle values.The CNN architectures are drawn from prior literature.
- Timing dataset: The timing dataset contains 50 three-dimensional microstructure images evaluated at four magnetic fields, for 200 data points.The data are split into 160 training, 20 validation, and 20 testing points.
- Timing limitation: CNN-2 cannot process microstructure images with Nd = 32, so its data-loading and training times are unavailable for that case.This limitation is specific to CNN-2's network architecture.
Integrated Gradient (IG) calculation
Integrated Gradients quantify how individual grain features contribute to a GNN prediction by comparing a real graph with a grain-size-zero baseline.
- Interpretability objective: Integrated Gradients quantify the importance of each physical feature in each grain for a given predicted magnetostriction.The method aggregates gradients along the straight path from baseline input to real input.
- Inputs and baseline: The real input is the microstructure graph G with external field Hx, while the baseline retains other features and sets grain size to zero.This baseline isolates the contribution of grain size to predicted magnetostriction.
- Feature preprocessing: Grain size and neighbor count are normalized within each microstructure graph to place them on a common scale while preserving node-to-node differences.The normalization uses the graph-specific mean and standard deviation of each feature.
- Attribution property: The sum of all Integrated Gradients equals the difference between the model outputs for the real and baseline inputs.The graph contains N grain nodes and five features per node in this analysis.
- Interpretation of scores: A positive or negative Integrated Gradient indicates that increasing a feature raises or lowers the prediction, respectively.Larger absolute Integrated Gradient values indicate greater feature importance.
Data Availability
The study’s 492-microstructure dataset supports its findings, with the dataset available through GitHub and other plotting data available from the authors upon request.
- The dataset of 492 microstructures supporting the study’s findings is available on GitHub.
- Other data supporting the paper’s plots are available from the corresponding authors upon reasonable request.