C4-equivariant flow matching on APD graphs generates polycrystalline microstructures with training-free guidance

Polycrystalline microstructure generation is formulated as flow matching over the compact geometric parameters of an APD, with each grain described by its position, power weight, and an ellipse-based anisotropy parametrisation. Existing APD generation methods prescribe selected grain statistics but do not learn a generative distribution over complete grain configurations or directly model spatial dependencies; this work learns the full APD parameter distribution. Trained on synthetic APDs with a model using three message-passing layers, a latent dimension, and nearest neighbours, for 100 epochs with the AdamW optimiser; generated samples can be rendered at arbitrary pixel resolution.

A graph-based C4-equivariant architecture models dependencies between grains and ensures consistent generative behaviour under rotations by multiples of 90 degrees. Rotational symmetry is built directly into the model rather than added through data augmentation or post-processing; message passing operates on invariant scalar features and predicts coefficients multiplying geometric bases with known transformation properties. The appendix gives a constructive derivation showing that for any rotation, rotating the initial state produces the correspondingly rotated generated APD.

A training-free guidance framework controls generated microstructures through user-defined differentiable objectives, imposing complex spatially varying global structure without retraining. Prior control over global characteristics was typically limited to statistics specified a priori by the sampling procedure; this framework allows application-specific objectives with separate RMS normalisation for position, anisotropy, and power-weight channels. Using the same pretrained model with different guidance functions, four morphology families are generated: 3D-printed layered structure, cast slab, weld, and heterogeneous lamella; target parameters and guidance strengths were selected manually for qualitative agreement with representative experimental microstructures.

The work targets researchers and engineers who need representative polycrystalline microstructures to model material behaviour, especially those who want to tailor global morphology to an application without retraining the model. The APD representation is independent of raster resolution, supports variable grain counts, and allows generated microstructures to be rendered at arbitrary pixel resolution; training-free guidance can impose spatially varying targets such as layer-wise orientation, cast-slab zoning, an inward weld growth field, and heterogeneous lamella bands. Guidance targets may be specified from domain knowledge, tuned against an application-specific validation criterion, or estimated from real data.

The model is currently trained only on synthetic APDs, so fidelity to higher-order experimental statistics remains unvalidated, and strong or conflicting guidance may move samples away from the learned distribution. Guidance objectives constrain only selected APD statistics, leaving unconstrained features to the pretrained flow, so the method provides approximate control rather than exact sampling from a conditional distribution. Target parameters and guidance strengths were selected manually for qualitative agreement and need not be optimal. Future work will evaluate and fine-tune on EBSD data, incorporate crystallographic orientation, extend to 3D microstructures, and explore physics-informed guidance objectives.