Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics

1 Introduction

• CG molecular dynamics aims to reach larger spatial and temporal scales than atomistic simulations, while requiring a force field that reproduces relevant molecular features.The paper frames these scales as useful for studying large protein-folding events and slow interactions between large molecules.Source: §1 · p. 2
• Variational force matching needs projected atomistic forces and has a high-variance estimator, while relative-entropy methods avoid force inputs but incur greater computational cost.Relative-entropy methods require iterative model sampling to estimate log-likelihood gradients.Source: §1 · p. 3
• Flow-matching avoids atomistic-force inputs while retaining sample efficiency, but its CG models have struggled to reproduce fine-grained thermodynamics and become unstable on larger proteins.It uses a two-stage approach, including a force-matching-like objective to extract a deterministic force field.Source: §1 · p. 3
• A single diffusion model, trained with a denoising loss, generates i.i.d. equilibrium CG samples and serves directly as a CG force field for simulations.The authors report improved performance across protein simulations, reproduction of the CG equilibrium distribution, and preservation of dynamics such as folding events.Source: Figure 1 · p. 2, §1 · p. 3

2 Background

• Coarse-graining maps atomistic coordinates to lower-dimensional CG coordinates, commonly by representing groups of atoms as beads.For molecular systems, the mapping is usually linear, with each bead coordinate formed from representative atom coordinates.Source: §2 · p. 3
• The projected atomistic equilibrium distribution defines a thermodynamically consistent effective CG potential of mean force, but computing its integral is usually intractable.The effective potential is defined up to an additive constant.Source: §2 · p. 4
• Variational force matching uses projected atomistic forces to approximate a thermodynamically consistent CG force field under constraints on the mapping.Noid et al. [9] give a consistency relation for linear mappings where each bead has at least one atom with a nonzero coefficient specific to that bead.Source: §2 · p. 4
• Relative-entropy minimization estimates the CG density and derives forces from its log density without requiring atomistic forces or constraints on the CG map.The density objective is equivalent to maximum likelihood for a finite sample from the target distribution.Source: §2 · p. 4
• Conventional relative-entropy training is impractical for higher-dimensional problems because it requires free-energy estimation or model sampling for gradient estimation.Flow-matching instead uses augmented normalizing flows and a second force-matching-like stage to distill a deterministic approximate CG force.Source: §2 · pp. 4–5

3 Diffusion Models for Coarse-Grained Molecular Dynamics

• DDPMs learn sampling by approximating the reverse of a forward diffusion process through denoising.Source: §3 · p. 5

3.1 Extracting Force Fields from Diffusion Models

• At low noise, the learned score approximates the CG equilibrium-distribution score, yielding Denoising Force Field forces from a diffusion model trained with denoising loss.Source: §3.1 · p. 6
• Unlike earlier score-force connections limited to local minima, training on equilibrium Boltzmann samples approximates forces across the full equilibrium distribution for stable, reliable CG MD.Source: §3.1 · p. 6

3.2 Molecular Dynamics with the Denoising Force Field

• The DFF drives CG MD through Langevin dynamics, with γ and T matched to atomistic data and noise level i left as the tunable hyperparameter.Source: §3.2 · p. 7
• Iterative diffusion and denoising at low noise approximates Brownian dynamics, with an implicitly defined simulation timestep.Source: §3.2 · p. 7

3.3 Denoising Force Field Architecture

• A conservative score is parameterized as an energy gradient; the authors report this constraint is crucial for stable simulations.Source: §3.3 · p. 7
• Pairwise bead differences enforce translation invariance, while data augmentation approximates SO(3) equivariance without reflection equivariance; validation relative squared error under rotations is < 10^-6.Source: §3.3 · p. 7
• A graph transformer implements the symmetry-constrained energy model, and no prior energy term was needed for stable simulations.Source: §3.3 · p. 8

4.1 Coarse-Grained Simulation — Alanine Dipeptide

• On alanine dipeptide, DFF sim. beats prior CG simulation methods, especially with little training data, while DFF i.i.d. approaches the train/test lower bound.The reported comparison uses Jensen-Shannon divergence between dihedral distributions; the figure averages four runs and shows 95% confidence intervals.Source: Figure 2 · p. 9, §4.1 · p. 9

4.2.1 Equilibrium Analysis

• Across fast-folders, DFF i.i.d. and DFF sim. outperform their respective baselines on equilibrium metrics, while DFF maps capture global off-diagonal structure better than flow-based models.The landscape similarity is weaker for Bba than for Chignolin and Villin; the authors hypothesize that β-sheets’ non-local contacts are harder to model.Source: Table 1 · p. 10, §4.2.1 · pp. 11–12, Figure 4 · p. 12

4.2 Coarse-Grained Simulation — Fast-folding Proteins

• The fast-folding evaluation covers Chignolin, Trp-cage, Bba, and Villin, plus 56-bead Protein G to assess scalability to a larger protein.The four fast-folders use 10, 20, 28, and 35 Cα beads, respectively; the paper notes flow-matching had scaling challenges on larger proteins.Source: §4.2 · p. 10

4.2.1 Equilibrium Analysis

• DFF sim. performance is sensitive to the network’s bias–variance trade-off and can decrease with more flexible networks.This contrasts with DFF i.i.d., which generally improves as network features or layers increase.Source: §4.2.1 · p. 12

4.2.2 Dynamics Analysis

• DFF sim. outperforms Flow-CGNet sim. across fast-folders on transition-probability metrics, and its Chignolin trajectory shows folding and unfolding transitions.Protein G is excluded from the transition comparison because Flow-CGNet sim. samples were unavailable for that protein.Source: Table 2 · p. 13, §4.2.2 · pp. 13–14, Figure 5 · p. 13

• Denoising diffusion models provide a new approach to coarse-grained molecular dynamics, motivated by connections among score-based models, force fields, and molecular dynamics.Source: §5 · p. 14
• The approach offers a simple training setup and improved performance and scalability compared with previous work.Source: §5 · p. 14
• Future directions include scaling to larger proteins, generalizing across systems, and combining force-agnostic training with force matching when force information is available.Source: §5 · p. 14

A.1 Relation between Score Function sθ(zi, i) and Noise Predicting Network ϵθ(zi, i)

• The noise-prediction loss is a reweighted denoising score-matching loss, and vice versa.Source: §A.1 · p. 19

A.2 Connecting Denoising Diffusion and the Brownian Dynamics

• At low noise, iterating diffusion and denoising approximates Brownian dynamics, with the first noise level β1 proportional to the simulation time step Δt.The denoising process corresponds to Brownian dynamics under a force field derived from the noise-prediction network.Source: §A.2 · pp. 19–20

B.1 Ablation on Conservative Forces

• Conservative forces are represented by taking the gradient of a scalar with respect to input coordinates, satisfying the stated path-independent-work property.Source: §B.1 · p. 21
• The conservative variant outperforms the non-conservative variant on most metrics for Chignolin and Trp-cage, with a particularly large gap for Chignolin simulated data.The Chignolin simulated-data gap is an order of magnitude for TIC JS and even larger for PWD JS.Source: §B.1 · p. 21, Figure S1 · p. 21, Table S1 · p. 22
• Increasing non-conservative network depth improves simulated dynamics for Chignolin, but performance remains well below the conservative variant; the i.i.d. generation gap narrows with larger networks.Source: §B.1 · p. 21

B.2 Ablation on Number of Hidden Features

• The capacity sweet spot is more sensitive for simulated dynamics than for i.i.d. generation, despite identical training settings across feature-count variants.Source: §B.2 · p. 22
• Simulated dynamics peak at 64 hidden features and deteriorate with larger networks, while i.i.d. generation improves as hidden features increase.Source: §B.2 · p. 22
• The authors identify robustness for higher-capacity networks and proteins with stronger non-local-contact dependence, including Bba and Protein G, as areas for improvement.Source: §B.2 · p. 22

B.3 Ablation on noise level.

• At smaller noise levels, q(zi) more closely resembles the reference distribution, but the Langevin simulation results involve a trade-off in the chosen noise level.Figure S3 compares the convolved data distribution with DFF simulations.Source: §B.3 · pp. 23–24, Figure S3 · p. 24

B.4 Rotation Equivariance through Data Augmentation

• Relative squared error stays below 10^-6 across all five proteins at extraction noise levels {20, 15, 5, 5, 5}, supporting data augmentation as sufficient for rotation equivariance in most practical use cases.The levels correspond to Chignolin, Trp-cage, Bba, Villin, and Protein G, respectively.Source: Table S3 · p. 24, §B.4 · p. 24

C.1 Architecture Details

• The network adapts a Graph Transformer: node inputs combine embeddings and noise level, edge inputs are vector differences, and a summed scalar output is differentiated with respect to z.The learnable parameters include the node embeddings.Source: §C.1 · p. 25

C.2 Coarse-graining Operator

• Coarse-graining selects backbone atoms for alanine dipeptide and Cα atoms for the fast-folding proteins.The systems overview depicts fine-grained and coarse-grained representations.Source: §C.2 · p. 25, Figure S4 · p. 26

C.3 The optimization objective

• Training samples noise levels from lower and upper ranges with equal probability, increasing the frequency of low-noise training and improving the extracted force field.Each range receives 50% probability.Source: §C.3 · p. 25

C.4 Alanine Dipeptide

• The alanine dipeptide dataset comprises four 500 ns simulations with 250K samples each; three simulations supply training and validation data, and one supplies testing data.Of the 750K samples from the three training-validation simulations, 500K are used for training and 250K for validation.Source: §C.4 · p. 26

C.4.1 Implementation details

• Training-set sizes of 10K, 20K, 50K, 100K, 200K, and 500K are used to evaluate the influence of dataset size on performance.Source: §C.4.1 · p. 26
• The network has two graph Transformer layers with 96 hidden features per layer and is trained with Adam at learning rate 3 · 10^-4, cosine decay to 1 · 10^-5, and batch size 1024.Experiments run for 1M iterations, with early stopping for training sets of 10K and 20K.Source: §C.4.1 · p. 26
• Langevin simulations run for 1M iterations at 2-femtosecond time steps, producing 400K samples across 100 parallel simulations.Samples are saved every 250 steps; temperature and friction match the reference data at 300 Kelvin and 1 ps^-1.Source: §C.4.1 · p. 26
• The noise level is cross-validated, with smaller training datasets yielding larger selected values; the table reports the selected levels by dataset size.In the idealized infinite-data and infinite-capacity case, the text says the smallest value, i = 1, would be chosen.Source: §C.4.1 · pp. 26–27, Table S4 · p. 27

C.5.1 Dataset details

• Fast-folding protein trajectories are Cα-only, sampled every 200 picoseconds, and randomly split 70-10-20% into training, validation, and test sets.The split uses the same seed as K¨ohler et al. 19.Source: §C.5.1 · p. 27, Table S5 · p. 27

C.5.2 Implementation details

• Diffusion training uses a cosine scheduler with 1000 noise levels and Adam; the appendix reports hyperparameters and per-system training speeds.Reported speeds range from 17 it/s for Chignolin on two V100 GPUs to 5.9 it/s for Protein G on four V100 GPUs.Source: §C.5.2 · p. 27, Table S6 · p. 28
• Langevin simulations run for 6 million steps, saving every 500 steps to produce 1.2M samples across 100 parallel simulations; noise levels are {20, 15, 5, 5, 5} for the five proteins.The listed order is Chignolin, Trp-cage, Bba, Villin, and Protein G.Source: §C.5.2 · p. 27

C.5.3 Extended results

• DFF i.i.d. captures off-diagonal contact distributions better than DFF simulation; DFF simulation still improves significantly on Flow-CGNet, and DFF i.i.d. is extremely close to the reference for Protein G.Protein G has no Flow-CGNet samples for comparison.Source: §C.5.3 · p. 28, Figure S5 · p. 30
• DFF simulation's bond distribution matches the diffused reference distribution qi(z), and the samples avoid unknown regions associated with bond dissociation.The bond distribution has the same mean as reference MD but broader variance attributed to diffusion noise.Source: §C.5.3 · pp. 28–29, Figure S6 · p. 31
• For sequence-distant atoms three or more residues apart, the small-distance tail does not approach zero, indicating no chemical-integrity violation such as backbone crossings.Near-zero distances would indicate van der Waals violations and could correspond to backbone crossings.Source: §C.5.3 · p. 29, Figure S6 · p. 31
• Figure S7 presents K-means clusters in 2D TIC space and transition-probability matrices for the reference distribution, Flow-CGNet, and DFF, with color intensity encoding probability.Source: §C.5.3 · p. 29, Figure S7 · p. 32