Scalable Autoregressive Deep Surrogates for Dendritic Microstructure Dynamics

TL;DR

This work tackles the computational bottleneck of phase-field simulations for dendritic microstructure by learning autoregressive deep surrogates (ADS) trained on short PF trajectories. It introduces a Scale-Invariant ConvNeXt–based ADS that predicts microstructure evolution from a single initial frame, enabling extrapolation in space and time without solving the PF PDEs at every step and achieving speed-ups over two orders of magnitude. The ADS reproduces key PF-derived metrics, including the tip-selection constant $\sigma^*$ (matching the analytical value), four-fold morphological symmetry, and primary dendrite spacing, across both isothermal growth and directional solidification scenarios in dilute Al-Cu alloys. This approach provides a practical route to integrating microstructure modeling into ICME workflows and suggests broad applicability to other pattern-forming processes in materials and energy systems.

Abstract

Microstructural pattern formation, such as dendrite growth, occurs widely in materials and energy systems, significantly influencing material properties and functional performance. While the phase-field method has emerged as a powerful computational tool for modeling microstructure dynamics, its high computational cost limits its integration into practical materials design workflows. Here, we introduce a machine-learning framework using autoregressive deep surrogates trained on short trajectories from quantitative phase-field simulations of alloy solidification in limited spatial domains. Once trained, these surrogates accurately predict dendritic evolution at scalable length and time scales, achieving a speed-up of more than two orders of magnitude. Demonstrations in isothermal growth and in directional solidification of a dilute Al-Cu alloy validate their ability to predict microstructure evolution. Quantitative comparisons with phase-field benchmarks further show excellent agreement in the tip-selection constant, morphological symmetry, and primary spacing evolution.

Figures (7)

• Figure 1: The workflow for using autoregressive deep surrogate (ADS) in microstructure prediction. It involves training the Scale-Invariant ConvNeXt (SI-ConvNeXt) model on short microstructural trajectories obtained from quantitative phase-field (PF) simulations conducted over limited spatial and temporal scales. Once trained, the surrogate model can be extrapolated to predict microstructure evolution at larger spatial and temporal scales.
• Figure 2: Predicting isothermal dendrite growth using ADS. (a) Training data are generated from $N_{\mathrm{train}} = 20$ PF simulations on a domain of size $1024 \times 1024\, (d_0^*)^2$ over a simulated time of $t_{\mathrm{tot}}^{\mathrm{train}} = 6 \times 10^4\, (d_0^*)^2 / D$. Once trained, the ADS is used to predict microstructure evolution at larger spatial and temporal scales. (b) Ground truth results from PF simulations of multiple dendrite growth. (c) Predicted microstructure using the trained ADS. In both (b) and (c), black contours represent $\phi = 0$, and the colormap denotes the $U$ field. (d)-(e) Spatial distributions of prediction errors $\Delta \phi$ and $\Delta U$ at $t = 2.16 \times 10^5\, (d_0^*)^2 / D$. (f) Spatially averaged mean-squared error $\mathrm{MSE}^i(t)$ for both $\phi$ and $U$ as a function of time. (g)-(k) Corresponding results for a single dendrite growth predicted using the same trained ADS model, analogous to (b)-(f).
• Figure 3: The normalized mean-squared error: (a) $\bigl\langle \mathrm{MSE}^\phi \bigr\rangle_t$ and (b) $\bigl\langle \mathrm{MSE}^U \bigr\rangle_t$ between the ground truth and predicted results for the $\phi$ and $U$ fields, respectively, plotted as a function of the extrapolation domain size $N_x$ (i.e., the number of grid points along the $x$-direction in a square simulation domain). Five surrogates are trained on the same dataset in a small domain size shown in Fig. \ref{['fig:dendrite']}(a). Each data point is the average of five ADS simulations using the five trained surrogates, and the error bars represent the standard deviation.
• Figure 4: (a) Comparison of the $\phi = 0$ contour of a single dendrite at $t = 1.96 \times 10^5\, (d_0^*)^2 / D$ from the GT and PD simulations. (b) Enlarged views of the dendrite tip regions from all dendrites in the GT and PD simulations, where the tip locations are aligned through rotation and translation to a common reference point. (c) Temporal evolution of the tip selection constant $\sigma^*$ for all dendrites in the GT and PD simulations. The black dashed line represents the analytical value $\sigma^* = 0.0396$, and the arrow indicates the time corresponding to the dendrite shown in panels (a) and (b).
• Figure 5: Time-averaged mean-squared errors $\langle \mathrm{MSE}^{i} \rangle_t$ and time-averaged symmetry errors $\langle E_{\mathrm{sym}}^{i} \rangle_t$ as functions of the number of training simulations ($N_{\mathrm{train}}$) and the ratio $R_t \equiv \Delta t / \Delta t_{\mathrm{PF}}$, where $\Delta t$ is the ADS prediction time step and $\Delta t_{\mathrm{PF}}$ is the time step of the PF finite-difference solver. Five surrogates are trained on the same dataset in a small domain size shown in Fig. \ref{['fig:dendrite']}(a). Each data point is the average of five ADS simulations using the five trained surrogates, and the error bars represent the standard deviation.