Grain growth without solving phase-field equations: VN law sets areas, a graph network rebuilds the microstructure, 137x faster on five benchmarks

Two-dimensional polycrystalline grain growth can now evolve grain areas alone and let a neural network turn those areas back into a spatial microstructure, without solving phase-field equations. Phase-field methods treat grain boundaries as diffuse interfaces and carry a heavy computational cost, which restricts the range of initial microstructures and kinetic conditions that can be explored. Across five initial microstructures compared at a common coarsening threshold of half the initial grain count, VN-GraphGG averaged 38.43 s of wall time against 5,247.01 s for the phase-field method, with an arithmetic mean case-wise ratio of 136.9.

Each step first converts the microstructure field into a graph, applies the VN law to update the area feature on every grain node, then lets a graph-convolution decoder map the graph back to a pixel-level field. Graph neural networks have mostly been used to learn whole evolution trajectories, where approximation errors accumulate when learned updates are applied iteratively. On 6,000 graph-field test pairs the decoder reached a mean per-field pixel error of 3.54%, boundary and adjacency F1 scores of 0.916 and 0.973, and a mean absolute area error of 2.04 pixels per grain.

Kinetic coefficients stay separate from the reconstruction model, so coefficients can be changed without retraining the network, and topological events are handled by reconstruction plus postprocessing. Statistical VN-based models evolve grain statistics only and do not supply a polycrystalline field for spatial analysis. In the abnormal-grain-growth study, sweeping the high-coefficient grain fraction at fixed contrast gave a peak occurrence fraction of 0.93 at an intermediate fraction, falling to 0.67 at 100%.

The method targets two-dimensional grain growth: the VN law prescribes each grain's area at every step, and a trained graph-to-field decoder restores the microstructure, so it suits simulations that need explicit grain-scale kinetic control and ensemble comparisons. The reconstruction model is trained once and then frozen, while kinetic coefficients can be adjusted without retraining. The authors propose extensions to larger domains, additional microstructure classes and experimental fields, and note that anisotropic boundary properties, solute drag, second-phase pinning and internal stress could enter through corresponding kinetic formulations and input features, with the generalized MacPherson–Srolovitz relation as a possible three-dimensional kinetic basis.

The method applies the VN law to the reconstructed microstructure at each step, which is an approximation; the VN law itself describes grain-scale kinetics but does not specify the evolving microstructure. In the phase-field benchmark, VN-GraphGG wall times are averaged over three repeated runs while each phase-field case is run once, and the phase-field time is the elapsed time to an observed stopping event rather than an interpolation between output states. Abnormal-grain-growth occurrence intervals come from bootstrap resampling clustered by initial field, with 30 simulations per condition from three selection seeds over ten initial fields.