L-MAU: A multivariate time-series network for predicting the Cahn-Hilliard microstructure evolutions via low-dimensional approaches

The phase-field model is a prominent mesoscopic computational framework for predicting diverse phase change processes. Recent advancements in machine learning algorithms offer the potential to accelerate simulations by data-driven dimensionality reduction techniques. Here, we detail our development of a multivariate spatiotemporal predicting network, termed the linearized Motion-Aware Unit (L-MAU), to predict phase-field microstructures at reduced dimensions precisely. We employ the numerical Cahn-Hilliard equation incorporating the Flory-Huggins free energy function and concentration-dependent mobility to generate training and validation data. This comprehensive dataset encompasses slow- and fast-coarsening systems exhibiting droplet-like and bicontinuous patterns. To address computational complexity, we propose three dimensionality reduction pipelines: (I) two-point correlation function (TPCF) with principal component analysis (PCA), (II) low-compression autoencoder (LCA) with PCA, and (III) high-compression autoencoder (HCA). Following the steps of transformation, prediction, and reconstruction, we rigorously evaluate the results using statistical descriptors, including the average TPCF, structure factor, domain growth, and the structural similarity index measure (SSIM), to ensure the fidelity of machine predictions. A comparative analysis reveals that the dual-stage LCA approach with 300 principal components delivers optimal outcomes with accurate evolution dynamics and reconstructed morphologies. Moreover, incorporating the physical mass-conservation constraint into this dual-stage configuration (designated as C-LCA) produces more coherent and compact low-dimensional representations, further enhancing spatiotemporal feature predictions. This novel dimensionality reduction approach enables high-fidelity predictions of phase-field evolutions with controllable errors, and the final recovered microstructures may improve numerical integration robustly to achieve desired later-stage phase separation morphologies.