Physically-Constrained Autoencoder-Assisted Bayesian Optimization for Refinement of High-Dimensional Defect-Sensitive Single Crystalline Structure

TL;DR

This work tackles defect-sensitive crystal-structure refinement by marrying a physically-constrained variational autoencoder (pc-VAE) with Bayesian optimization (BO) and Sparse Axis-Aligned Subspace BO (SAASBO) to efficiently navigate high-dimensional parameter spaces. The pc-VAE enforces scale invariance and non-negativity in learned diffraction representations, enabling a robust latent-space surrogate for optimization. Across 2D, 4D, and 8D explorations in Ho$_2$Ti$_2$O$_7$, pc-VAE–assisted BO and SAASBO demonstrate faster convergence and improved alignment with experimental structure factors relative to traditional Rietveld refinement. The approach offers a path toward defect-aware, data-efficient refinement that can accelerate discovery in defect-sensitive materials and frustrated magnet systems, with potential extensions to multi-objective and neutron diffraction data.

Abstract

Physical properties and functionalities of materials are dictated by global crystal structures as well as local defects. To establish a structure-property relationship, not only the crystallographic symmetry but also quantitative knowledge about defects are required. Here we present a hybrid Machine Learning framework that integrates a physically-constrained variational-autoencoder (pcVAE) with different Bayesian Optimization (BO) methods to systematically accelerate and improve crystal structure refinement with resolution of defects. We chose the pyrochlore structured Ho2Ti2O7 as a model system and employed the GSAS2 package for benchmarking crystallographic parameters from Rietveld refinement. However, the function space of these material systems is highly nonlinear, which limits optimizers like traditional Rietveld refinement, into trapping at local minima. Also, these naive methods don't provide an extensive learning about the overall function space, which is essential for large space, large time consuming explorations to identify various potential regions of interest. Thus, we present the approach of exploring the high Dimensional structure parameters of defect sensitive systems via pretrained pcVAE assisted BO and Sparse Axis Aligned BO. The pcVAE projects high-Dimensional diffraction data consisting of thousands of independently measured diffraction orders into a lowD latent space while enforcing scaling invariance and physical relevance. Then via BO methods, we aim to minimize the L2 norm based chisq errors in the real and latent spaces separately between experimental and simulated diffraction patterns, thereby steering the refinement towards potential optimum crystal structure parameters. We investigated and compared the results among different pcVAE assisted BO, non pcVAE assisted BO, and Rietveld refinement.

Figures (8)

• Figure 1: Schematic of the crystal structure, single-crystal X-ray diffraction geometry, and associated reciprocal-space construction used in this work. In the two crystal models shown as insets, Ho, Ti, and O atoms are represented by dark blue, light blue, and red balls, separately. The crystal model on the left highlights the two inter-penetrating tetrahedra networks formed by Ho-Ho and Ti-Ti connections, while the crystal model on the right highlights the oxygen environments and the TiO${}_6$ cage (light blue octahedra). The cubic lattice vectors are indicated by the $\bf a, b, c$ vectors forming the coordinate system shown as red, blue, and green arrows. A monochromatic X-ray beam illuminates the single crystal, generating Bragg-diffracted beams that intersect the area detector, where each spot corresponds to a reciprocal lattice vector $\mathbf{G} = h\mathbf{a}^* + k\mathbf{b}^* + l\mathbf{c}^*$. The magnitude of $\mathbf{G}$ determines the real-space lattice spacing $d = 2\pi / \lvert \mathbf{G} \rvert$. The inset at the bottom right illustrates the scattering plane, showing the relationship between the incident and diffracted beams, the scattering angle $2\theta$, and the projected interplanar spacing $d$.
• Figure 2: Proposed architecture of the pc-VAE with physical constraints and scale invariance. Here, the physical constrained is defined as the intensity of the reconstructed diffraction are non-negative.
• Figure 3: Proposed architecture of the pc-VAE assisted BO and SAASBO exploration for crystalline structure refinement of pyrochlore model Ho$_2$Ti$_2$O$_7$. The yellow arrows are the additional steps for the pc-VAE BO while the green arrows are the steps for traditional BO. The orange arrows are the common steps for the BO and the pc-VAE-BO.
• Figure 4: Performance of the implementation of scaling invariance, physical constraints and hyperparameter optimization of pc-VAE. Panels (a) and (b): The performance comparison of VAE without and with the scaling invariance respectively. No physical constraints had been imposed in this case. Panels (c) and (d): The performance comparison for VAE trained without and with the physical constraint for non-negativity of diffraction intensities respectively. No scaling invariance have been imposed in this case. The black circled region highlights the physical constraint violation of the reconstruction in (c) and no such violation in (d). For panels (a)-(d), two examples of reconstruction of the diffraction pattern have been shown out of 25000 training data. Panels (e), (f) and (g): Optimized hyperparameters such as latent dimension, learning rate and training epochs respectively, for the training process of pc-VAE with imposing scaling invariance and physical constraints.
• Figure 5: Convergence of BO and pc-VAE-BO over the 4-D structure parameter space. In each of the panels (a) and (b), top and bottom rows represent the maps of GP mean and GP uncertainty respectively, while panels (a) and (b) represent BO and pc-VAE-BO, respectively. The table on the top right indicates the structure parameters placed on the X and Y axes for each of the 6 columns in panels (a) and (b). Panels (c) and (d) present the convergence plots of $\chi^2$ and $\chi_l^2$ for BO and pc-VAE-BO respectively.