Riemannian Flow Models with Reinforcement Learning for Molecular Crystal Structure Prediction

Flow-based generative models can efficiently produce candidate structures for crystal structure prediction (CSP), but their pretrained objectives do not directly optimize downstream target recovery. Reinforcement-learning post-training offers a flexible solution, yet existing approaches rely primarily on energy rewards and coordinate-only stochastic policies. Predicted energy does not identify the reference polymorph, while reward-driven concentration can reduce the candidate coverage required for Top-N recovery. We introduce CrystalGRPO, a CSP-aligned post-training framework that extends existing ODE-to-SDE policy constructions to the joint coordinate--lattice state. CrystalGRPO combines MACE-predicted energy with a StructureMatcher-based recovery score and provides two operating modes: CrystalGRPO-Q, which prioritizes single-draw recovery, and CrystalGRPO-C, which combines full-trajectory reference regularization with a coverage-aware group advantage to preserve finite-budget target recovery. Across MP-20 and MPTS-52 with PXRDGen and OMatG backbones, both variants reduce one- and twenty-sample RMSE relative to coordinate-only reinforcement in all four backbone--dataset settings. CrystalGRPO-Q consistently improves Top-1, whereas CrystalGRPO-C achieves a higher Top-20 across all settings.

Crystal structure prediction (CSP) is fundamental to computational materials discovery. Generative models including CDVAE, DiffCSP, FlowMM, and CrystalFlow learn stable-crystal distributions directly, but diffusion and flow-matching inference requires tens to thousands of sequential network evaluations per candidate. We introduce uFlowCSP, a MeanFlow-based CSP model that learns the average, rather than instantaneous, probability-flow velocity. It generates a complete structure in one to five evaluations, delivering 5x-58x faster inference with equal or better performance. A chemistry- and symmetry-aware Transformer uses canonical atom ordering, global composition, and per-token chemistry embeddings. A coarse crystal-system token is used only during training; it provides additive gains, particularly improving space-group agreement despite being absent at inference, which remains formula-only. On MP-20 with 20 candidates per target, one step matches CrystalFlow (78.38% vs. 78.34%) with 100x fewer evaluations and about 10x lower wall-clock time. Five steps reach 83.64%, exceeding CrystalFlow (78.34% at 2,000 evaluations) and DiffCSP (77.93% at about 20,000), while using 20x fewer evaluations. uFlowCSP generates 10,000 structures in 0.39-1.31 minutes, versus 6.5 for CrystalFlow and 76.1 for DiffCSP. Under CSPBench's energy-ranked top-five structure-and-space-group criterion, five-step uFlowCSP reaches 72%/72%/65% structure, space-group, and consensus match rates. CrystalFlow reaches 78%/73%/68% at 100 steps but falls to 49%/32%/31% at five. Thus, uFlowCSP improves accuracy per network evaluation, not merely peak accuracy.

Progress in AI-driven crystal materials science has so far been carried by narrow architectures purpose-built for individual tasks -- graph neural networks for property prediction, diffusion and flow-matching models for crystal generation -- each excelling within its niche yet unable to act as a shared backbone across the full spectrum of materials problems. Generative large language models offer a fundamentally different paradigm, in which structural representation, quantitative prediction, and structure-activity reasoning can be unified within one model, but the materials community has yet to see this paradigm realized at a level competitive with established narrow specialists. Here we present MatMind, a generative foundation model purpose-built for crystal materials science under this paradigm, developed through the coordinated activation of structure-activity knowledge and physics-informed feedback within a progressive training framework -- combining structure-activity knowledge injection, a dual-head architecture that jointly trains language reasoning and numerical regression in a shared representation space, and multi-objective physics-informed reinforcement learning over stability, novelty, and structural diversity. Across three task families, MatMind attains the lowest mean absolute error on energy above hull, bulk modulus, and band gap -- surpassing graph neural network predictors purpose-built for these tasks -- reaches an S.U.N. rate of 65.3% on unconditional crystal generation, and achieves a comparable multiplicative improvement on magnetization-density-conditioned generation, where only 21 positive samples exist within over 600000 training entries. By matching or surpassing narrow specialists on their own ground while operating within a single unified model, MatMind shows that the LLM-based paradigm can serve as a viable backbone for crystal materials science going forward.