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Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models
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Proposes an energy-based diffusion model regularized with a Fokker–Planck residual loss and a time-based mixture of experts so that score-derived forces give simulations consistent with iid diffusion samples on dipeptides, Chignolin and BBA.
Problem
Energy-based interpretation of diffusion-model scores (score at t=0 as forces) is inconsistent with the distribution recovered by diffusion sampling, which breaks molecular dynamics simulation from learned forces, especially for coarse-grained biomolecules.
Approach
Add a weak-form Fokker–Planck residual regularization on the log-density of a conservative (gradient-of-energy) score network, combined with a mixture of time-interval experts so that only the small-diffusion-time expert is energy-based and regularized.
Key results
- On alanine dipeptide, the Mixture model reduces simulation PMF error to 0.325 ± 0.113 versus 1.047 ± 0.924 for the plain Diffusion baseline, with iid PMF roughly unchanged.
- Fokker-Planck regularization yields consistent iid and simulated ensembles for Chignolin and BBA, where Two For One shows noisy simulations and BBA separates into unphysical modes.
- The mixture of experts cuts iid sampling time by more than 50% on Chignolin (43 min to 14 min) and BBA (12h57 to 5h56).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use three conservative diffusion experts instead of one Diffusion model (ablation) | Diffusion · Alanine dipeptide (sim) | JS divergence: 0.0695 ± 0.0517 -> 0.0264 ± 0.0085 | Helped |
| use three conservative diffusion experts instead of one Diffusion model (ablation) | Diffusion · Alanine dipeptide (sim) | PMF error: 1.047 ± 0.924 -> 0.325 ± 0.113 | Helped |
| use three conservative diffusion experts instead of one Diffusion model (ablation) | Diffusion · Alanine dipeptide (iid) | JS divergence: 0.0081 ± 0.0003 -> 0.0079 ± 0.0003 | Helped |
| use three conservative diffusion experts instead of one Diffusion model (ablation) | Diffusion · Alanine dipeptide (iid) | PMF error: 0.095 ± 0.003 -> 0.093 ± 0.007 | Helped |
| use wider layers (ablation) | Diffusion · Alanine dipeptide (sim) | JS divergence: 0.0695 ± 0.0517 -> 0.0406 ± 0.0236 | Helped |
| use more layers (ablation) | Diffusion · Alanine dipeptide (sim) | PMF error: 1.047 ± 0.924 -> 0.478 ± 0.038 | Helped |
| add Fokker–Planck regularization (ablation) | energy-based diffusion models · molecular systems | sampling-simulation consistency | Helped |
| use a mixture of experts trained on smaller diffusion-time intervals (ablation) | Diffusion · dipeptides | simulation PMF error | Helped |
| combine mixture of experts with Fokker–Planck regularization (ablation) | Fokker–Planck model · dipeptides | PMF error | Helped |
| evaluate the model at a larger diffusion timestep during simulation (ablation) | Diffusion · molecular simulations | structural fidelity | Hurt |
| use a conservative energy-based parameterization instead of a direct score-based model (ablation) | diffusion model · molecular simulation | simulation stability | Helped |
| use a conservative energy-based parameterization instead of a direct score-based model (ablation) | diffusion model · iid sampling | sample quality | No clear effect |
Surprising findings
- Mixture of experts improves consistency through a different mechanism than Fokker-Planck regularization: its Fokker-Planck error at t=0 stays comparable to unregularized models.
- Score-based (non-conservative) models could not be trained stably for simulation without a conservative parameterization.
Limitations
- Fokker-Planck residual evaluation requires multiple forward passes per training step, making training computationally expensive.
- Results are primarily empirical; perfect alignment between diffusion sampling and Langevin simulation may be unattainable without limiting expressivity.
- Evaluated on relatively small coarse-grained systems; larger molecular systems left to future work.
Experimental setup
Datasets: alanine dipeptide implicit-solvent MD (bgmol) Chignolin and BBA explicit-solvent MD (Lindorff-Larsen/D. E. Shaw style fast-folding data) 2AA dipeptide dataset (timewarp)
Benchmarks: alanine dipeptide Chignolin BBA 2AA dipeptides test set (15 dipeptides)
Metrics: PMF error Jensen-Shannon divergence free energy surface comparison Fokker-Planck residual
Baselines: Two For One Diffusion (conservative, t=1e-5) Transferable Boltzmann Generator
Builds on: denoising score matching variational diffusion models Two For One Transferable Boltzmann Generator Langevin dynamics
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