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FlowMM: Generating Materials with Riemannian Flow Matching

Benjamin Kurt Miller, Ricky T. Q. Chen, Anuroop Sriram, Brandon M Wood

TL;DR

<3-5 sentence high-level summary>FlowMM tackles the challenging space of periodic crystal design by casting CSP and DNG as symmetry-aware, flow-based generative problems on a Riemannian manifold representing crystals. It generalizes Riemannian Flow Matching to enforce translation, rotation, permutation, and periodic boundary conditions, using a carefully crafted base distribution and a binary atom-type encoding for DNG. The method delivers state-of-the-art or competitive performance with far fewer integration steps than diffusion-based approaches, and it validates generated structures via quantum-chemistry stability assessments, demonstrating practical efficiency gains. This framework offers a scalable, invariant-aware paradigm for rapid materials discovery with directly measurable stability metrics.

Abstract

Crystalline materials are a fundamental component in next-generation technologies, yet modeling their distribution presents unique computational challenges. Of the plausible arrangements of atoms in a periodic lattice only a vanishingly small percentage are thermodynamically stable, which is a key indicator of the materials that can be experimentally realized. Two fundamental tasks in this area are to (a) predict the stable crystal structure of a known composition of elements and (b) propose novel compositions along with their stable structures. We present FlowMM, a pair of generative models that achieve state-of-the-art performance on both tasks while being more efficient and more flexible than competing methods. We generalize Riemannian Flow Matching to suit the symmetries inherent to crystals: translation, rotation, permutation, and periodic boundary conditions. Our framework enables the freedom to choose the flow base distributions, drastically simplifying the problem of learning crystal structures compared with diffusion models. In addition to standard benchmarks, we validate FlowMM's generated structures with quantum chemistry calculations, demonstrating that it is about 3x more efficient, in terms of integration steps, at finding stable materials compared to previous open methods.

FlowMM: Generating Materials with Riemannian Flow Matching

TL;DR

<3-5 sentence high-level summary>FlowMM tackles the challenging space of periodic crystal design by casting CSP and DNG as symmetry-aware, flow-based generative problems on a Riemannian manifold representing crystals. It generalizes Riemannian Flow Matching to enforce translation, rotation, permutation, and periodic boundary conditions, using a carefully crafted base distribution and a binary atom-type encoding for DNG. The method delivers state-of-the-art or competitive performance with far fewer integration steps than diffusion-based approaches, and it validates generated structures via quantum-chemistry stability assessments, demonstrating practical efficiency gains. This framework offers a scalable, invariant-aware paradigm for rapid materials discovery with directly measurable stability metrics.

Abstract

Crystalline materials are a fundamental component in next-generation technologies, yet modeling their distribution presents unique computational challenges. Of the plausible arrangements of atoms in a periodic lattice only a vanishingly small percentage are thermodynamically stable, which is a key indicator of the materials that can be experimentally realized. Two fundamental tasks in this area are to (a) predict the stable crystal structure of a known composition of elements and (b) propose novel compositions along with their stable structures. We present FlowMM, a pair of generative models that achieve state-of-the-art performance on both tasks while being more efficient and more flexible than competing methods. We generalize Riemannian Flow Matching to suit the symmetries inherent to crystals: translation, rotation, permutation, and periodic boundary conditions. Our framework enables the freedom to choose the flow base distributions, drastically simplifying the problem of learning crystal structures compared with diffusion models. In addition to standard benchmarks, we validate FlowMM's generated structures with quantum chemistry calculations, demonstrating that it is about 3x more efficient, in terms of integration steps, at finding stable materials compared to previous open methods.
Paper Structure (61 sections, 2 theorems, 26 equations, 9 figures, 8 tables)

This paper contains 61 sections, 2 theorems, 26 equations, 9 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Preliminaries
  5. Representing crystals and their symmetries
  6. Representation of a crystal
  7. Equivariance
  8. Invariant density
  9. Symmetries of crystals
  10. Representing atomic types
  11. Learning distributions with Flow Matching
  12. Riemannian Manifolds (Abridged)
  13. Probability paths on flat manifolds
  14. Flow Matching
  15. Crystalline Solids
  16. ...and 46 more sections

Key Result

Theorem 4.1

For pairwise $G$-invariant conditional probability path $p_t(x|x_1)$, meaning $p_t(g \cdot x| g \cdot x_1)=p_t(x|x_1) \;\forall g\in G,\;x,x_1\in \mathcal{C}$, the construction in eqn:unconditional_probability_path_is_invariant defines a $G$-invariant marginal distribution $p_t(x)$.

Figures (9)

  • Figure 1: A conceptual representation of the evolution from base distribution to target distribution, according the vector field learned by our model. We model a joint distribution over lattice parameters, periodic fractional coordinates, and a binary representation of atom type. The highlights include symmetry-aware geodesic paths and a base distribution that directly produces plausible lattices. Note that coordinates and atom types are merely depicted in 2d for clarity.
  • Figure 2: Match rate as a function of number of integration steps on MP-20. FlowMM achieves a higher maximum match rate than DiffCSP overall, and does so $\sim$ 450 steps before DiffCSP. Results with Inference Anti-Annealing ablated are in Appendix \ref{['appendix:results_continued']}.
  • Figure 3: The distribution of number of unique elements per material, or $N$-ary, for the MP-20 distribution and the generative models. FlowMM matches the MP-20 distribution closest, while CDVAE and DiffCSP generate too many materials with $N$-ary $\geq$ 5.
  • Figure 4: Histogram comparing the distribution of $E^{hull}$ computed after relaxation with DFT for generative models DiffCSP and CDVAE with our proposed FlowMM on the DNG task. After relaxation on for all models, FlowMM generates lower energy structures compared to CDVAE and is competitive with DiffCSP.
  • Figure 5: Three symmetry actions are shown above; all of these actions would alter only the representation of the crystal, while leaving its chemical properties intact. (top) Rotation of a lattice formed by a unit cell. (mid) Translation of fractional coordinates within a unit cell. (bot) Permutation of atomic index. Since these images are two-dimensional, they do not capture the symmetry of a three-dimensional crystal. Furthermore, there are additional symmetries that are not represented in these pictures.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof