Graph neural networks for an accurate and interpretable prediction of the properties of polycrystalline materials

Polycrystalline materials’ macroscopic properties depend on both the lattice structure within grains and microstructural attributes such as grain size, shape, orientation, and adjacency. Predicting structure–property relationships using physics-based simulations can be intractable for large, complex polycrystals, and accurate physics-based models may be difficult to develop in systems with complex compositions or under extreme conditions. Existing machine learning approaches for microstructure–property prediction largely fall into statistical-descriptor-based and image-based models that extract correlations via low-dimensional embeddings, yet they do not explicitly encode grain adjacency and thus cannot directly capture interactions among neighboring grains that critically influence macroscopic behavior. The study proposes a graph neural network framework that represents a polycrystalline microstructure as a graph of grains and their adjacencies, enabling message passing to model inter-grain interactions and thereby improving prediction accuracy and interpretability for effective properties, demonstrated on magnetostriction of Terfenol-D.

Two broad classes of models are discussed. Statistical-descriptor-based models represent microstructure via correlation functions (e.g., two-point correlation) linking features at spatial locations to probabilities. Image-based models (e.g., 3D CNNs) take voxelized microstructure images as input and learn embeddings (feature maps) automatically. Both approaches correlate features across space but lack explicit storage of adjacency relations among grains, preventing direct modeling of neighbor interactions. Prior GNNs have been applied to molecules and crystals but not polycrystals. This gap motivates a graph-based representation that captures grain-level features and adjacencies to encode physically meaningful interactions in polycrystals.

Microstructure graph construction: Each grain is a node with a feature vector containing five components: three Euler angles (α, β, γ; zxz sequence) describing orientation, grain size (number of voxels), and number of neighboring grains. The adjacency matrix A encodes grain contacts (Aij=1 if grains i and j are neighbors; 0 otherwise). The graph input is G=(F, A). GNN model: A graph convolutional network (GCN) performs message passing through layers (MPLs) with update function F(n+1)=σ(D−1/2 Â D−1/2 F(n) W(n)), where Â= A+I, D is the degree matrix, σ is ReLU, and W(n) are trainable weights. This operation mixes node features with neighbors to encode inter-grain interactions. The final embedding {X} is concatenated and, together with the applied magnetic field Hx, passed to a fully connected layer to regress the effective magnetostriction λeff. The fully connected layers compute vi=σ(Σj wij uj + bi). Dataset generation and labels: Using Dream.3D, 492 distinct 3D polycrystalline microstructures were generated with grains per microstructure ranging from 12 to 297 and including textured and non-textured cases. Phase-field simulations on 64×64×64 grids computed local magnetization and magnetostriction under magnetic fields along x (Hx), yielding effective magnetostriction λeff by volume-averaging λxx. For each microstructure, 4–5 values of Hx were simulated, producing 2287 data points of the form [(G, Hx), λeff]. Phase-field modeling includes exchange, stray, external, anisotropy, and elastic energies; LLG dynamics with material constants for Terfenol-D; periodic boundaries; and Fourier spectral methods for fields and elastic equilibrium. Training, validation, and testing: The 2287 points were split into 10 subsets: 8 for training (1831 points), 1 for validation (228), 1 for testing (228). Eighty-one GNN hyperparameter combinations were trained; the model with lowest validation MARE was selected. Optimized hyperparameters: W(1): 5×60, W(2): 60×3; hidden units Nh1=1024, Nh2=128; batch size=32; epochs=2000; learning rate=1e−4. Data ablation studies randomly selected subsets of k microstructures (k from 32 to 392) with tenfold cross-validation. Computational efficiency benchmarks: Three 3D CNN baselines (from prior literature) were trained on 50 microstructures and 4 fields (200 points) with identical batch size and epochs, on one Tesla P100 GPU. Training time (data loading + training) was compared as a function of voxel count Na^3 (32–128), with Ngrain fixed at 300 for each image in the CNN vs GNN comparison, and as a function of Ngrain (300–4700) for GNN. Interpretability via Integrated Gradients (IG): IG quantifies the contribution of each feature of each grain to predicted λeff. A baseline graph sets grain size features to zero while keeping other features (α, β, γ, neighbors) equal to the target graph, enabling isolation of size contributions. IG integrates gradients along the straight-line path from baseline to real input; larger absolute IG indicates higher importance. Correlation between normalized grain size and its absolute IG value was analyzed per microstructure to compare the roles of size vs orientation.

- Accuracy: On the independent test set, the GNN achieved a macro average relative error (MARE) of 8.24% for λeff prediction across 492 microstructures (2287 data points). - Data efficiency: In ablation, average MARE decreased from ~16% to ~12% as the number of microstructures reached 72; with further data the error decreased slowly and saturated at ~9%. The standard deviation of MARE decreased and stabilized as dataset size grew, indicating improved stability. - Computational efficiency: Predicting 1 data point via phase-field simulation takes ~5 hours on 16 CPU cores; the trained GNN predicts 228 data points in ~0.2 s. Compared to 3D CNNs, GNN training time is largely independent of voxel count (given fixed Ngrain) and scales with number of grains. For images with 120^3 voxels and 300 grains, CNN-2 requires ~35× longer training time than GNN. GNN trained with up to 4700 grains fits within a single Tesla P100 GPU memory, with ~13 h total training time for 160 data points. - Interpretability: IG analysis on 492 microstructures shows that grain orientation generally plays a more important role than grain size in determining effective magnetostriction, evidenced by weak correlations between normalized grain size and its absolute IG across most graphs. In a specific 12-node graph, a strong correlation (R²>0.9) indicated size dominance for that case, but the broader dataset trends favored orientation importance. - Generality across microstructures: A single GNN architecture with fixed hyperparameters achieved low error across diverse microstructures with 12–297 grains and varying textures.

Representing polycrystalline microstructures as graphs enables explicit modeling of inter-grain interactions via message passing, aligning with the physical reality that macroscopic properties arise from both grain-intrinsic features and neighbor interactions. The GNN embedding fuses grain orientation, size, and connectivity, leading to accurate λeff predictions with low relative error and good data efficiency. The IG-based attribution provides grain- and feature-level interpretability, revealing that, in general, orientation dominates size in controlling magnetostriction, consistent with physical understanding that magnetostriction varies strongly with crystallographic orientation. Computationally, node-based graph representations scale with number of grains rather than voxels, offering significant efficiency advantages over voxel-based CNNs and enabling application to large 3D microstructures. These results demonstrate that graph-based learning can effectively capture the microstructure–property link while providing physically meaningful explanations.

The study introduces a microstructure-graph-based GNN that jointly encodes grain features and adjacency to predict effective properties of polycrystals. Applied to Terfenol-D magnetostriction, the model achieves 8.24% MARE on test data and maintains ~10% error even with relatively small datasets, offering rapid inference and strong interpretability via Integrated Gradients. The approach scales efficiently to large microstructures and outperforms voxel-based CNNs in training efficiency. Future directions include incorporating grain boundary features via edge attributes, representing other mesoscale defects with additional nodes, tailoring graph complexity and update functions to specific materials and properties, and exploring transfer learning by pretraining on related datasets. The framework can be extended to other tasks such as processing–microstructure and microstructure evolution predictions.

A single universal GNN with fixed graph inputs and update functions is unlikely to generalize across all polycrystalline materials and properties; appropriate features (including potential edge attributes for grain boundaries) and architectures must be selected based on the target system’s physics. The design of baseline inputs and post-processing for IG must align with physical principles to yield meaningful attributions. Computational resource limits for very large numbers of grains still apply, though less severely than for voxel-based CNNs. The study’s datasets are generated via simulations with specific material parameters and conditions, which may affect generalizability to experimental systems without domain adaptation or transfer learning.