A Physics-Regulated Neural Framework for Learning 3D Grain Growth Dynamics

Machine Learning (ML) models for grain growth prediction are typically trained on idealized synthetic data, yet practical applications require generalization to conditions outside the training distribution. This study evaluated the Out-Of-Distribution (OOD) generalization capability of the trained model from our previous study across three test cases, including experimental microstructures, microstructures characterized by a bimodal grain size distribution, and abnormal grain growth. To further probe whether physics-informed architectural design could improve robustness under these different conditions, a boundary-masked attention mechanism was proposed specifically for grain growth, constraining attention to grain boundary pixels. Both the baseline and the proposed physics-informed attention model were evaluated without retraining or fine-tuning on the OOD data. Both models successfully generalized to all three test cases, yet the boundary-masked attention mechanism provided substantial improvements, with the most notable gains for microstructures characterized by a bimodal grain size distribution, where Structural Similarity Index Measure (SSIM) improved from \num{0.6221} to \num{0.7609} and mean grain size ($\overline{R}$) error decreased from \SI{8.75}{\percent} to \SI{3.57}{\percent}. The attention heatmap analysis revealed that the boundary-masked attention model learned to concentrate attention on large grain boundaries in a manner consistent with curvature-driven grain growth physics, emerging from training without being explicitly encoded into the architecture. These results indicate that models trained on synthetic data can generalize to diverse OOD conditions without retraining, and that physics-informed attention may improve accuracy when the boundary morphology matches the training domain.

Understanding and predicting microstructure evolution is central to materials design, yet purely data-driven spatiotemporal learning models often suffer from limited physical consistency and degraded long-term prediction accuracy. In this work, we introduce a physics-guided fully convolutional spatiotemporal learning framework for microstructure evolution prediction. Unlike prior self-supervised approaches, the proposed method explicitly incorporates governing physical equations into the training objective, thereby encouraging the learned dynamics to remain consistent with known thermodynamic and kinetic laws. This physics-guided formulation improves predictive accuracy, long-horizon stability, and robustness across spatial resolutions and temporal prediction settings. Extensive experiments for spinodal decomposition demonstrate that incorporating physics-guided residual regularization leads to more faithful reproduction of microstructural morphology, statistics, and evolution trends compared with purely data-driven baselines. The proposed framework preserves the scalability and computational efficiency of fully convolutional architectures while bridging the gap between high-fidelity physics-based simulations and data-driven surrogate modeling, offering a reliable and efficient surrogate-modeling step toward digital-twin-enabled microstructure evolution prediction.

Neural surrogate models for physical simulations are trained on discretized samples of continuous domains, where the induced empirical measure leads to uneven supervision, biasing optimization and causing spatial inconsistencies in physical fidelity. To mitigate this measure-induced bias, we propose M$^3$ (Multi-scale Morton Measure), a scalable framework that balances training measures by partitioning space according to physical variation and allocating supervision across multiple scales. Applied to three industrial-scale datasets with diverse discretizations, M$^3$ consistently improves predictions in the continuous physical domain, achieving up to 4.7$\times$ lower error in large-scale volumetric cases. These gains persist under aggressive subsampling (160M $\rightarrow$ 16M $\rightarrow$ 1.6M points), where M$^3$-trained models outperform those trained on higher-resolution data, reducing physics-weighted relative $L_2$ error by 3--4$\times$ and the corresponding MSE by up to 13$\times$. These results highlight data distribution as a key factor in operator learning and position M$^3$ as a scalable, data-efficient approach for physically consistent modeling. Code is available at https://github.com/PhysDataRefine/M3.