NeurIPS 2025

Consistent Sampling and Simulation: Molecular
Dynamics with Energy-Based Diffusion Models
Michael Plainer† 1,2,3,4
Hao Wu† 5
Leon Klein 1
Stephan Günnemann 6
Frank Noé† 1,7,8
1Freie Universität Berlin
2Zuse School ELIZA
3Technische Universität Berlin
4Berlin Institute for the Foundations of Learning and Data 5Shanghai Jiao Tong University
6Technische Universität München
7Rice University
8Microsoft Research AI4Science
Abstract
In recent years, diffusion models trained on equilibrium molecular distributions
have proven effective for sampling biomolecules. Beyond direct sampling, the
score of such a model can also be used to derive the forces that act on molecular
systems. However, while classical diffusion sampling usually recovers the training
distribution, the corresponding energy-based interpretation of the learned score is
often inconsistent with this distribution, even for low-dimensional toy systems. We
trace this inconsistency to inaccuracies of the learned score at very small diffusion
timesteps, where the model must capture the correct evolution of the data distribu-
tion. In this regime, diffusion models fail to satisfy the Fokker–Planck equation,
which governs the evolution of the score. We interpret this deviation as one source
of the observed inconsistencies and propose an energy-based diffusion model with a
Fokker–Planck-derived regularization term to enforce consistency. We demonstrate
our approach by sampling and simulating multiple biomolecular systems, including
fast-folding proteins, and by introducing a state-of-the-art transferable Boltzmann
emulator for dipeptides that supports simulation and achieves improved consistency
and efficient sampling. Our code, model weights, and self-contained JAX and
PyTorch notebooks are available at https://github.com/noegroup/ScoreMD.
1
Introduction
Understanding biochemical systems requires modeling not only static molecular structures but also
their temporal evolution and interactions. Molecular dynamics (MD) si...