$C_4$-Equivariant Flow Matching on Anisotropic Power-Diagram Graphs for Microstructure Generation

Dawid Lipinski; Jixiang Qing; Henry Moss

cs.LGOct 8, 2026

$C_4$ -Equivariant Flow Matching on Anisotropic Power-Diagram Graphs for Microstructure Generation

Authors: Dawid Lipinski, Jixiang Qing, Henry Moss

Organizations: School of Mathematical Sciences Lancaster University Lancaster, LA1 4YF, United Kingdom

Abstract

Acquiring realistic microstructure data through Electron Backscatter Diffraction (EBSD) is costly and time consuming, often relying on specialised equipment. As microstructures strongly influence material properties, generating realistic samples is essential for modelling the behaviour of polycrystalline materials. We introduce a generative model for synthesising realistic polycrystalline microstructures using flow matching and graph neural networks. By representing microstructures as anisotropic power diagrams, our model learns a compact geometric parametrisation and can render generated samples at arbitrary pixel resolution. A $C_4$ -equivariant architecture incorporates rotational symmetry directly into the model, ensuring that rotations of the input noise produce corresponding rotations of the generated microstructure. We also demonstrate how training-free guidance can be used to generate complex microstructures, based on user defined objective function. In particular, we generate microstructures resembling a copper weld, cast metal slab, 3D-printed stainless steel and heterogeneous lamella titanium.

Figures & tables

Panel of Figure 1 — Figure 1 : Unguided (left) and guided (right) samples generated by our model from the same initial noise. Each panel shows the ellipse representation on the left and the corresponding APD tessellation on the right. Guidance encourages four horizontal bands with alternating grain orientations of $+45^{\circ}$ and $-45^{\circ}$ , resembling a microstructure pattern in 3D-printed stainless steel.

Figure 2 : Each row shows training-free guidance of synthetic polycrystalline microstructures. The first four columns show independent samples generated by the same pretrained model, while the rightmost column presents representative experimental microstructures for qualitative comparison. The real examples are taken from (top to bottom): Buze et al. [2] , Balit et al. [1] , Reed-Hill [8] , Savolainen et al. [10] , Wu et al. [13]

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Objective $\mathcal{J}_{k}$	$\lambda_{k,x}$	$\lambda_{k,a}$	$\lambda_{k,w}$
Layer orientation	$(0,0.03)$	$18$	$0$
Cast shape	$(0.02,0)$	$16$	$0$
Cast relative weight	$(0.02,0)$	$0$	$0.30$
Cast generator density	$(0.25,0)$	$0$	$0$
Weld orientation	$0.02$	$18$	$0$
Lamella topology	$(0.04,0.26)$	$0$	$0$

Appendix

Table 1: Guidance strengths used for Figure 2 . An ordered pair gives horizontal and vertical position strengths; a scalar position strength is applied to both components. Since each gradient is RMS-normalised before scaling, the values specify correction magnitudes in normalised model coordinates.

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One-Step Flow Matching for Generative Modeling of Path-Dependent Physical Fields

Yijing Zhou, Jasmin Jelovica

aDepartment of Mechanical Engineering, The University of British Columbia, Vancouver, Canada · bDepartment of Civil Engineering, The University of British Columbia, Vancouver, Canada

Flow Matching Structural Mechanics PDE Surrogate Modeling One-Step Generative Modeling Flow-Based Generative Modeling

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Physical simulations for intricate geometries with path-dependent constitutive models face difficulties due to the enormous computational cost they require. Recently, the emergence of generative AI models, which succeed in image and video synthesis tasks, has provided a promise to further improve simulations. Although U-Net-based denoising diffusion probabilistic models (DDPMs) have been adopted for elastic stress field generation, they typically require hundreds of sampling steps, and applications of generative models to path-dependent, e.g. plastic, stress fields remain very limited. In this work, we propose a novel flow matching (FM) model based on a transformer backbone for high-resolution path-dependent stress field generation with stochastic loading-unloading paths and geometry. The proposed model operates within the latent space of a variational autoencoder (VAE) and formulates the simulation of plastic fields as a video synthesis task, directly generating the stress fields across all time steps. Meanwhile, we design a non-Gaussian source distribution for flow matching, such that crossings among conditional transport paths are reduced during training. This enables our model to generate satisfactory samples in one step without relying on distillation. In addition, we introduce token-level loading embeddings and two auxiliary networks to further enhance the model performance in path-dependent simulation. The results demonstrate that, even with a limited training dataset, our model can accurately generate high-resolution path-dependent fields. It is much more computationally efficient than finite element analysis, providing a speedup of 6 to 7 times over FEM on CPUs and approximately two orders of magnitude speedup on consumer-grade GPUs.

Abstract

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