Deep Learning-Driven Prediction of Microstructure Evolution via Latent Space Interpolation

TL;DR

This work tackles the high computational cost of phase-field simulations for microstructure evolution by introducing a CVAE-based surrogate conditioned on alloy composition, augmented with cubic spline interpolation in latent space and SLERP for time evolution. Trained on 6,300 images across nine binary spinodal decomposition compositions with $c_{avg}$ in $[0.27,0.48]$, the CVAE learns a compact latent representation and can generate intermediate microstructures for target compositions via latent-space interpolation. The combination of cubic spline-based composition interpolation and SLERP-based temporal evolution yields microstructures that closely match phase-field simulations in both temporal coarsening and spatial statistics (e.g., two-point autocorrelation), while reducing generation time from minutes to a fraction of that time. This approach offers a scalable surrogate for rapid materials design and composition optimization, enabling efficient exploration of large composition spaces without repeatedly solving the governing PDEs.

Abstract

Phase-field models accurately simulate microstructure evolution, but their dependence on solving complex differential equations makes them computationally expensive. This work achieves a significant acceleration via a novel deep learning-based framework, utilizing a Conditional Variational Autoencoder (CVAE) coupled with Cubic Spline Interpolation and Spherical Linear Interpolation (SLERP). We demonstrate the method for binary spinodal decomposition by predicting microstructure evolution for intermediate alloy compositions from a limited set of training compositions. First, using microstructures from phase-field simulations of binary spinodal decomposition, we train the CVAE, which learns compact latent representations that encode essential morphological features. Next, we use cubic spline interpolation in the latent space to predict microstructures for any unknown composition. Finally, SLERP ensures smooth morphological evolution with time that closely resembles coarsening. The predicted microstructures exhibit high visual and statistical similarity to phase-field simulations. This framework offers a scalable and efficient surrogate model for microstructure evolution, enabling accelerated materials design and composition optimization.

Figures (7)

• Figure 1: Workflow of the Conditional Variational Autoencoder (CVAE) framework for modeling microstructure evolution. The CVAE takes as input a set of microstructure images $x_1, x_2, ..., x_n$ along with their corresponding covariate information (cavg). The encoder maps the inputs to a latent space $z = \mu + \sigma \odot \varepsilon$. The dotted red arrows represent the cubic spline interpolation workflow in the latent space, where representations for the targeted composition value are generated, which are then decoded to produce synthetic microstructures. The purple arrows show the selection process of latent vectors of images with the smallest and largest feature size (white shapes) and their injection back into the encoder. The green arrows depict the SLERP (Spherical Linear Interpolation) workflow, which operates between latent vectors corresponding to the microstructures with the largest and smallest feature size (white shapes) to generate a sequence of microstructures showing gradual and realistic morphological transitions of the targeted composition value.
• Figure 2: Schematic representation of the Conditional Variational Autoencoder (CVAE) architecture. The encoder maps the input $x$ and covariate information $c$ into a latent distribution parameterized by mean $\mu$ and standard deviation $\sigma$. A latent variable $z$ is sampled and decoded back into the reconstructed output $\hat{x}$ while conditioning on $c$.
• Figure 3: Visualization of cubic spline interpolation applied to the conditional label vectors in a 9-dimensional label space. Each subplot illustrates the smooth variation of one label dimension as a function of the interpolation parameter 'alpha', constructed from known one-hot encoded composition vectors corresponding to alloy compositions between 0.27 and 0.48. The peaks in each dimension reflect dominant contributions. These splines enable continuous and physically meaningful transitions between known compositions, thereby facilitating smooth label-guided image synthesis within the CVAE framework.
• Figure 4: Illustration of Spherical Linear Interpolation (SLERP). Given two latent vectors $z_1$ and $z_2$, SLERP computes an intermediate point along the geodesic path on the unit hypersphere. The interpolation factor $\alpha$ determines the position of the generated latent vector, ensuring smooth and consistent transitions in the latent space.
• Figure 5: Comparison of original and CVAE-generated microstructure evolution for two composition intervals: (a) $0.27 \rightarrow 0.39$ and (b) $0.40 \rightarrow 0.46$.