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

Dawid Lipinski, Jixiang Qing, Henry Moss

arXiv:2610.11549v1cs.LG

TL;DR

EBSD data are costly to acquire, while microstructure generation needs to preserve grain geometry and support control of global patterns. The paper uses flow matching over APD parameters with a C4-equivariant graph architecture and training-free guidance. It demonstrates guided examples resembling several material structures, while noting that fidelity to higher-order experimental statistics remains unvalidated.

  • Problem

    EBSD measurements are costly, and existing approaches have limited control over global microstructure characteristics such as collective grain alignment and spatial patterns.

    Source: §1 · p. 1

  • Method

    The model applies flow matching to compact APD grain parameters with a C4-equivariant graph architecture, and uses differentiable objectives for training-free guidance.

    Source: §1 · p. 2, §3 · p. 3, §2 · p. 3

  • Results

    Guidance generated examples qualitatively resembling copper welds, cast metal slabs, 3D-printed stainless steel, and heterogeneous lamella titanium.

    Source: §4 · p. 4

  • Takeaways & Limitations

    The APD parametrisation is independent of raster resolution, and the model’s updates preserve corresponding generated rotations for 90°-rotated initial states.

    Source: §3 · p. 3, C C4-equivariant model updates · p. 12

    The model is trained only on synthetic APDs, so fidelity to higher-order experimental statistics remains unvalidated.

    Source: §5 · p. 5

Abstract

from arXiv · show

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.

1 Introduction

Realistic microstructures matter for modelling material behaviour, but EBSD data are costly and difficult to use at scale, while existing methods offer limited control over global structure. The paper addresses these gaps with compact APD flow matching, a C4-equivariant graph model, and training-free guidance.

1 Introduction

  • EBSD datasets are costly, limited, noisy, and too large for complete-sample processing, while many approaches provide limited control over global grain alignment and spatial patterns.The motivation is to generate representative microstructures for modelling material behaviour and studying structure–property relationships.Source: §1 · p. 1
  • Existing APD methods capture anisotropic grains compactly but do not learn distributions over complete configurations or directly model spatial dependencies.Their global control is typically limited to statistics prescribed by the sampling procedure.Source: §1 · p. 1
  • The model uses flow matching over compact APD parameters, supporting variable grain counts and rendering at arbitrary pixel resolution.Each grain is represented by its position and ellipse-based anisotropy parametrisation.Source: §1 · p. 2
  • A C4-equivariant graph architecture models dependencies between grains, while training-free differentiable-objective guidance imposes complex global structure without retraining.Examples are qualitatively inspired by copper welds, cast metal slabs, 3D-printed stainless steel, and heterogeneous lamella titanium.Source: §1 · p. 2

2 Background

The background introduces anisotropic power diagrams as compact tessellations whose grain regions are determined by seed locations, anisotropy, and weights. It then frames flow matching and objective-based guidance as the generation and control mechanisms.

2 Background

  • An anisotropic power diagram partitions the domain into regions determined by each seed’s location, anisotropic distance, and weight, with weights able to match target region areas.The anisotropy matrix is symmetric positive-definite; for fixed seed positions, anisotropy matrices, and target areas, weights can be obtained through optimal transport.Source: §2 · p. 2
  • Flow matching transports samples from a simple source distribution to the target distribution by integrating a learned vector field, which here predicts APD parameters with a graph neural network.The stated conditional path is linear between source and target samples.Source: §2 · pp. 2–3
  • Training-free guidance biases generation toward a differentiable objective by modifying the velocity using a terminal-state estimate, while keeping model parameters fixed.The guidance strength is controlled by β and its time schedule by s(t).Source: §2 · p. 3

3 Equivariant flow matching on APDs

The method represents microstructures compactly as APD grain parameters and uses a graph neural network with C4-equivariant updates. This makes generation resolution-independent while ensuring that 90° rotations of the initial state yield corresponding rotations of generated APDs.

3 Equivariant flow matching on APDs

  • The APD representation stores each grain’s seed position, power weight, and two anisotropy parameters, independent of raster resolution and regularising small pixel-level segmentation artefacts.The representation addresses the computational cost of modelling images that may contain millions of pixels while retaining their underlying geometric structure.Source: §3 · p. 3
  • The equivariant graph architecture connects each grain to its k nearest neighbours and processes invariant edge features through time-conditioned message-passing layers.Its output combines invariant scalar coefficients with geometric bases that transform equivariantly under 90° rotations.Source: §3 · p. 3
  • For any 90° rotation, the predicted velocity rotates correspondingly, so rotating the initial state produces the correspondingly rotated generated APD.The update is designed to preserve C4 equivariance through invariant features and equivariant geometric bases.Source: §3 · p. 3

4 Experiments

Experiments train a graph model on synthetic APDs and apply training-free objectives to produce several targeted microstructure patterns. The examples are qualitative, with target parameters and guidance strengths selected manually.

4 Experiments

  • The model was trained for 100 epochs on 12,000 synthetic APDs containing between 100 and 500 grains, using three message-passing layers and 16 nearest neighbours.The latent dimension was 256 and the optimiser was AdamW.Source: §4 · p. 3
  • Guidance produces examples resembling 3D-printed stainless steel, cast slabs, welds, and heterogeneous lamella titanium from the same pretrained model.The objectives target selected statistics, leaving the pretrained flow to determine remaining local geometry; Figure 2 places independent generated samples alongside experimental references for qualitative comparison.Source: §4 · p. 4, Figure 2 · p. 4
  • Guidance uses a cosine ramp with no guidance during the first 10% of the trajectory, and the neighbourhood graph is recomputed as generator positions evolve.Target parameters and guidance strengths were selected manually for qualitative agreement with representative experimental microstructures, and need not be optimal.Source: §4 · p. 5

5 Discussion

The model is trained only on synthetic APDs, leaving its fidelity to higher-order experimental statistics unvalidated. Strong or conflicting guidance may also move samples away from the learned distribution.

5 Discussion

  • Training uses only synthetic APDs, so fidelity to higher-order experimental statistics remains unvalidated.Source: §5 · p. 5
  • Strong or conflicting guidance may move generated samples away from the learned distribution.Source: §5 · p. 5
  • Future work will evaluate and fine-tune on EBSD data, add crystallographic orientation, extend to 3D, and explore physics-informed guidance objectives.Source: §5 · p. 5

A Guidance objectives and sampling details

The appendix specifies training-free guidance and sampling procedures for producing Figure 2’s examples. Guidance acts through objective gradients and selected APD statistics, while unconstrained features remain determined by the pretrained flow.

Guidance and sampling

  • In the guided velocity, gradients pass through both the objective and vθ in the terminal estimate, while each objective–channel pair is RMS-normalised separately before the corrections are summed.The schedule has a 10% unguided warm-up followed by a cosine ramp. All figure samples use N = 300 generators, k = 16 nearest neighbours, and K = 60 Euler steps.Source: A Guidance objectives and sampling… · p. 7

Layered 3D-printed structure

  • Layer guidance alternates its target direction across six horizontal layers and suppresses ambiguous orientations at interfaces.Detaching the confidence prevents grains from reducing the loss merely by moving toward an interface.Source: A Guidance objectives and sampling… · p. 8

Cast slab

  • Cast-slab guidance combines chill, columnar, and core zones with spatially varying shape, relative-weight, and generator-density targets.The columnar target is horizontal, with an axis ratio increasing from 2.2 to 9.2; target generator fractions are 0.34, 0.48, and 0.18 for the three zones.Source: A Guidance objectives and sampling… · p. 8

Weld structure

Heterogeneous-lamella structure

  • Lamella guidance places the coarse-grain population in four wavy bands while separating coarse and fine weight populations without fixing grain labels.Its shape term weakly aligns coarse grains with local band tangents and drives fine grains toward equiaxed shapes.Source: A Guidance objectives and sampling… · p. 9

Scope and sampling

  • The objectives constrain selected APD statistics, leaving unconstrained features to the pretrained flow; strong or conflicting objectives may leave its learned data manifold.Reference images are used only for qualitative comparison and do not enter any objective.Source: A Guidance objectives and sampling… · pp. 6, 9

B Anisotropy matrix parametrisation

The anisotropy parametrisation predicts unconstrained scalars while preserving valid anisotropy matrices by construction. A unit-determinant constraint separates grain shape from size and gives the parameters a simple rotation rule.

B Anisotropy matrix parametrisation

C C4-equivariant model updates

The model constructs updates from rotation-invariant scalar features and geometric bases with known transformation laws. This makes its velocity field C4-equivariant, so rotating the initial state yields the corresponding rotation of the generated APD.

C C4-equivariant model updates

  • The square is centred in transformed coordinates so rotations act linearly; positions and anisotropy transform under C4 while scalar APD weights remain invariant.Source: C C4-equivariant model updates · pp. 10–11
  • The network predicts invariant scalar coefficients from C4-invariant features, then multiplies them by geometric bases that transform equivariantly.This separates learned contribution magnitudes from their prescribed rotation behaviour.Source: C C4-equivariant model updates · pp. 11–12
  • Message passing uses invariant edge features and an invariant attention score to weight neighbouring grains while preserving equivariance.Source: C C4-equivariant model updates · p. 11
  • The vector field satisfies vθ(Rz, t) = R vθ(z, t) for rotations in C4, and integrating it preserves this equivariance in generated APDs.Rotating the initial state by a multiple of 90° produces the corresponding rotation of the generated APD.Source: C C4-equivariant model updates · pp. 11–12
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