Curvature-Aware Flow Matching for Molecular Structure Generation
Abstract
Generative models produce three-dimensional molecular structures with high geometric accuracy, yet they are trained on distributions of atomic coordinates without explicit access to the underlying physics. The potential energy surface (PES) describes how the energy changes around a molecular geometry, and the RMSD between a generated structure and its reference captures this local landscape only indirectly. Supervising the PES during training could close this gap, but conformers and transition states (TSs) are stationary points at which the forces vanish, so first-order quantities carry little information about them. What remains is the local curvature, encoded in the Hessian of the PES. Quantum-chemical Hessians are too expensive to be available at the scale required for training, but recently proposed machine-learned predictors make curvature inexpensive enough to supervise directly. We introduce CurvFM, a flow matching (FM) formulation whose training objective constrains the curvature at the generated structure. A frozen Hessian prediction model is evaluated at the generated endpoint and at the reference structure, which removes the need for quantum-chemical Hessians and keeps the residual cheap enough to impose at every training step. Across TS and conformer generation, CurvFM substantially improves physical fidelity at unchanged global structural accuracy, reducing the median deviation in maximum force by 63% on GEOM-QM9 and raising the fraction of valid TSs by up to 10% across three reaction datasets.
Related
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BibTeX
@misc{darouich2026curvatureaware,
title = {{Curvature-Aware Flow Matching for Molecular Structure Generation}},
author = {Samir Darouich and Juliane Geisler and Jacob W. Toney and Tanja Bien and Johannes Kästner and Heather J. Kulik and Mathias Niepert},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34592},
eprint = {2609.34592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34592},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature-Aware Flow Matching for Molecular Structure Generation
AU - Samir Darouich
AU - Juliane Geisler
AU - Jacob W. Toney
AU - Tanja Bien
AU - Johannes Kästner
AU - Heather J. Kulik
AU - Mathias Niepert
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34592
UR - https://arxiv.org/abs/2609.34592
AB - Generative models produce three-dimensional molecular structures with high geometric accuracy, yet they are trained on distributions of atomic coordinates without explicit access to the underlying physics. The potential energy surface (PES) describes how the energy changes around a molecular geometry, and the RMSD between a generated structure and its reference captures this local landscape only indirectly. Supervising the PES during training could close this gap, but conformers and transition states (TSs) are stationary points at which the forces vanish, so first-order quantities carry little information about them. What remains is the local curvature, encoded in the Hessian of the PES. Quantum-chemical Hessians are too expensive to be available at the scale required for training, but recently proposed machine-learned predictors make curvature inexpensive enough to supervise directly. We introduce CurvFM, a flow matching (FM) formulation whose training objective constrains the curvature at the generated structure. A frozen Hessian prediction model is evaluated at the generated endpoint and at the reference structure, which removes the need for quantum-chemical Hessians and keeps the residual cheap enough to impose at every training step. Across TS and conformer generation, CurvFM substantially improves physical fidelity at unchanged global structural accuracy, reducing the median deviation in maximum force by 63% on GEOM-QM9 and raising the fraction of valid TSs by up to 10% across three reaction datasets.
ER -