TexFlow: learning crystallographic texture on SO(3) manifold via normalizing flows

Abstract

Crystallographic texture is an important microstructural feature that statistically describes the orientations of grains in polycrystalline materials. Conventional texture analysis relies on the orientation distribution functions and pole figures, which are essential for interpreting the texture of a given material specimen but do not, by themselves, provide a direct framework for predictive modeling of process-structure-property linkages. Data- and machine learning-based approaches offer a means to address this gap; however, a key challenge lies in developing representations and learning frameworks that account for the manifold structure of rotations. In particular, many existing machine learning techniques are designed for Euclidean data and therefore are not readily applicable to rotations in the non-Euclidean group $\text{SO}(3)$. Here we introduce TexFlow, a conditional normalizing flow framework for texture modeling that learns a probability distribution over rotations in $\text{SO}(3)$ by construction. Rotations are encoded in a six-dimensional Gram-Schmidt orthogonalization representation which is then mapped back to valid rotations $\mathbf{R}\in\text{SO}(3)$. The framework supports conditional probabilistic modeling of texture $p(\mathbf{R}|c)$, where $c$ may represent processing parameters like temperature or deformation state. To complement conventional texture-analysis visualizations, such as pole figures and sections of the orientation distribution function, we further introduce Hopf-fibration visualizations as a compact global representation of texture. TexFlow is validated on both unconditional and conditional experimental and simulated texture datasets spanning diverse material systems. We show that TexFlow can accurately capture multimodal textures and their conditional evolution, providing a data-driven framework for establishing process-texture linkages.