Excited to share our latest preprint: 𝗟𝗲𝗮𝗿𝗻𝗶𝗻𝗴 𝗛𝗮𝗺𝗶𝗹𝘁𝗼𝗻𝗶𝗮𝗻 𝗙𝗹𝗼𝘄 𝗠𝗮𝗽𝘀: 𝗠𝗲𝗮𝗻 𝗙𝗹𝗼𝘄 𝗖𝗼𝗻𝘀𝗶𝘀𝘁𝗲𝗻𝗰𝘆 𝗳𝗼𝗿 𝗟𝗮𝗿𝗴𝗲-𝗧𝗶𝗺𝗲𝘀𝘁𝗲𝗽 𝗠𝗼𝗹𝗲𝗰𝘂𝗹𝗮𝗿 𝗗𝘆𝗻𝗮𝗺𝗶𝗰𝘀 🎉

Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models 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 ...

(7/n) Check out our paper and code:
Paper: arxiv.org/abs/2506.17139
Code + models: github.com/noegroup/Sco...

And also, our self-contained notebooks!
Colab (JAX):
colab.research.google.com/drive/1r3DGO...
Colab (PyTorch):
colab.research.google.com/drive/1rbcND...

#NeurIPS2025 #Diffusion #MD

(6/n)

Done with a brilliant team: Hao Wu, Leon Klein, Stephan Günnemann, and @franknoe.bsky.social .

Comparison of equilibrium distributions obtained by iid sampling and Langevin simulation (sim) across different systems and methods. While classical iid sampling recovers the reference equilibrium distribution, performing simulation with the learned score reveals inconsistencies when models are not trained with Fokker-Planck regularization, i.e., p(x) != p_0(x). Regularized models achieve consistent behavior across systems.

(5/n) With this, we can run coarse-grained Langevin dynamics directly, without the need for any priors or force labels.

This works across biomolecular systems including fast-folding proteins like Chignolin and BBA.

Here is a comparison with and without our regularization:

A model trained with Fokker–Planck regularization is self-consistent, and aligns the learned score at t = 0 with the distribution recovered by diffusion sampling.

(4/n) Our solution:

We train an energy-based diffusion model and regularize it to satisfy the Fokker–Planck equation.

This enforces consistency between:

- The density recovered via denoising
- The potential energy learned at t = 0

Result: the same model can be used for sampling AND simulation.

(3/n) The root issue is that at very small diffusion times, diffusion models are inaccurate.

The loss is large, and the models violate the Fokker-Planck equation, meaning the evolution of the model’s density and its score disagree.

When that happens, the recovered energy 𝑼(x) is not meaningful.

Training diffusion models on a 2D toy example reveals inconsistencies. While classical iid diffusion sampling (i.e., denoising) correctly reproduces both modes, evaluating the score at t = 0 to estimate the unnormalized density yields a third mode and an incorrect mass distribution. Such a diffusion model would produce incorrect dynamics while producing correct samples.

(2/n) The problem: classical diffusion models learn scores that reproduce equilibrium samples, but the corresponding energy-based parameterization is not consistent.

So if you try to use the learned energy to derive forces, the dynamics are wrong, even if the samples themselves look fine.

(1/n) Can diffusion models simulate molecular dynamics instead of just generating independent samples?

In our NeurIPS 2025 paper, we train energy-based diffusion models that can do both:
- Generate independent samples
- Learn the underlying potential 𝑼

🧵👇
Paper: arxiv.org/abs/2506.17139