CrysPFlow: Physics-Supervised Flow Matching for Crystal Relaxation

Crystal structure relaxation seeks stable, low-energy structures from unrelaxed configurations. First-principles methods achieve high accuracy through repeated energy and force evaluations, but incur substantial computational cost, while direct learning-based approaches largely rely on relaxed-structure supervision and underuse the local directional information encoded by the potential-energy surface (PES). We introduce CrySPFlow, a physics-supervised Flow Matching framework that combines geometric endpoint supervision with PES-derived force feedback during training. To make physical supervision reflect the model-induced relaxation dynamics, we propagate terminal force feedback through a short differentiable manifold rollout. We further introduce an endpoint-preserving gradient injection rule that incorporates physical descent information while preserving the geometric objective’s first-order descent rate in raw-gradient space. The interatomic potential is used only during training, so inference remains purely flow-based without additional potential evaluations. On $\mathrm{SrTi}{\mathrm{O}}_3$, CrySPFlow reduces coordinate and bond-length MAEs by 40.6% and 48.5%, respectively, relative to the strongest baseline for each metric. On X–Mn–O, it reduces coordinate MAE by 9.26%, while experiments with adapted FlowMM, RG-VFM, and CrystalFlow backbones show consistent benefits of physics supervision across the tested architectures.