Material Microstructure Design Using VAE-Regression with Multimodal Prior

TL;DR

The paper tackles forward and inverse structure–property prediction in materials science by coupling a variational autoencoder with a regression model through a conditional prior $p(z|c)$, enabling direct inverse inference without optimization loops. It introduces a multi-modal Gaussian mixture prior for $p(z|c)$ and replaces pixel-level reconstruction with a style loss based on Gram statistics from a pretrained network, enhancing robustness to microstructure texture. Experiments on 3D microstructures show forward predictions are on par with state-of-the-art methods, while inverse inference yields multiple plausible microstructures corresponding to target properties, with substantially reduced computation when refined by local physics-based optimization. The approach scales to multi-property targets, maintaining strong forward accuracy and improving inverse inference by capturing distinct morphologies per mixture component, thereby enabling efficient, physics-informed design in materials systems. Overall, the method provides a practical, data-driven route to direct inverse design and rapid design-space exploration in computational materials science.

Abstract

We propose a variational autoencoder (VAE)-based model for building forward and inverse structure-property linkages, a problem of paramount importance in computational materials science. Our model systematically combines VAE with regression, linking the two models through a two-level prior conditioned on the regression variables. The regression loss is optimized jointly with the reconstruction loss of the variational autoencoder, learning microstructure features relevant for property prediction and reconstruction. The resultant model can be used for both forward and inverse prediction i.e., for predicting the properties of a given microstructure as well as for predicting the microstructure required to obtain given properties. Since the inverse problem is ill-posed (one-to-many), we derive the objective function using a multi-modal Gaussian mixture prior enabling the model to infer multiple microstructures for a target set of properties. We show that for forward prediction, our model is as accurate as state-of-the-art forward-only models. Additionally, our method enables direct inverse inference. We show that the microstructures inferred using our model achieve desired properties reasonably accurately, avoiding the need for expensive optimization loops.

Figures (11)

• Figure 1: Architecture : VAE-Regression with Style Loss.
• Figure 2: Two microstructures with $C_{11} \approxeq 25GPa$. The volume fraction of black phase in (a) is 74.49% and in (b) is 26.02%, as shown by the green line in (c)
• Figure 3: (a) Inverse inference for target $C_{11} = 30GPa$. Top row shows real microstructures with the target $C_{11}$. The microstructures inferred using a uni-modal Gaussian prior (middle row) tend to be like an average, whereas the multi-modal Gaussian mixture prior learns the multiple possible solutions under separate mixture components (last row). (b) Evaluation of inverse inference through FEM simulations to get the achieved $C_{11}$ and compute the absolute % error between target and achieved $C_{11}$. The multi-modal prior learns multiple solutions under separate components unlike the uni-modal Gaussian prior, leading to more accurate inference (error within 20% in most cases).
• Figure 4: Optimization based inverse inference (a) For target $C_{11} = 35 GPa$, real microstructures (top row), those obtained through optimization by starting at 5 random initial points of which two were distinct (middle row), and at the points inferred by our method (last row). (b) Visualization in latent space. Optimization starting from the points inferred by our method (red) converges to the orange points in 10 iterations, leading to high-quality solutions (c) The absolute % error for a range of target $C_{11}$ values. When starting at random points, at least 5 searches, 200 iterations each, are needed to achieve mean error $<$10%. Starting at points inferred by our method, this is achieved in just 10 iterations
• Figure 5: 3D Style loss from 2D slices - (a)The observed microstructure, (b)Reconstruction by matching 51 slices in x direction, (c)&(d) Center slices of (b) in x and y directions, and (e)&(f)Center slices of a reconstruction by matching slices in all directions. Features in y direction are left unmatched in (d), as compared to (f).