Learning data-efficient coarse-grained molecular dynamics from forces and noise

Domain: chemistry
Published: 2026-03-15
Editorial outcome: minor_revision

Abstract
Molecular dynamics (MD) simulations are essential for elucidating biomolecular function, yet the computational cost of all-atom models often limits their reach. Machine-learned coarse-grained (MLCG) models offer a solution by simplifying the representation while maintaining near-atomistic accuracy. However, the training of MLCG models currently requires vast amounts of force-labeled sample conformations from reference atomistic MD. Here, we overcome this limitation by unifying the training of MLCG models with the principles of generative diffusion models. We demonstrate that accurate high-dimensional distributions of molecular ensembles can be recovered by integrating traditional force-matching with denoising objectives. This framework enables the construction of physically consistent and stable force fields while reducing atomistic data requirements by up to two orders of magnitude. Validated across diverse protein folds and scales, our work establishes a bridge between molecular dynamics simulation and modern generative learning, substantially lowering the computational cost of constructing accurate MLCG models and broadening their applicability to large biomolecular systems.

Introduction
Atomistic molecular dynamics (MD) can, in principle, connect the microscopic motion of a large number of atoms to emergent properties1–4. However, despite decades of development, its application to large biomolecular systems is hampered by its computational cost. As atomistic MD computes the interactions of all biomolecular and solvent atoms and advances simulation time by a few femtoseconds per step, it is often computationally prohibitive to simulate timescales relevant for biomolecular dynamics. This has led to the creation of coarse-grained (CG) MD models, which use far fewer effective atoms and can employ effectively longer simulation time-steps, resulting in large speedups5–11. For example, while atomistic MD could model a solvated protein by explicitly representing each atom in the protein and surrounding water, CG MD could do so by considering only the atoms in the protein backbone6.

Unfortunately, while atomistic MD has achieved high accuracy12,13, CG MD has struggled to do so9–11. MD propagates each particle in time according to the forces on each atom—the negative derivatives of a potential energy function, which is generally called a force-field. Especially for biomolecular systems, atomistic force fields have been improved over decades using a combination of quantum mechanical calculations and experimental data. As a result, despite limitations in their functional form, modern atomistic force fields have reached a level of high predictiveness in terms of metastable structures and free energy differences, if sufficient sampling is used13–18. However, CG force fields are still far away from their atomistic counterparts in generalization and accuracy. Recent progress has suggested that machine learning (ML) may be central to achieving a comparable level of performance19–23, as ML can learn crucial many-body interactions which are often missing from traditional CG force-fields24,25.

Many methods exist to represent and train CG force-fields8–10. Training methods typically pursue either a top–down approach, in which the values of macroscopic quantities are matched10,11, or a bottom-up approach that aims to reproduce the detailed behavior present in a corresponding atomistic model8,9. Here we focus on the bottom-up approach following multiscale coarse-graining (also known as “variational force-matching”)26–30. In terms of the CG force-field representation, it is nontrivial how to design the functional form of the force-field itself to be sufficiently expressive yet practical for a given application. Recent attempts have borrowed function representations from ML21 (e.g., neural networks31), an approach that has seen preliminary success in modeling biomolecules19,20,22–24,32–39. However, learning the corresponding force-field has required either substantial computation or a significant amount of data21. The resources necessary for the creation of machine-learned coarse-grained (MLCG) force fields represent a significant barrier to the development of the next generation of CG models, which could rival the accuracy of atomistic MD.

Here, we present a strategy for training bottom-up neural-network CG force fields using reference atomistic MD simulation data that is computationally efficient and reduces data requirements by a factor of 100. This is made possible by unifying denoising score matching40–42 (the key technology underlying denoising diffusion models (DDMs)43–45, a popular type of ML generative model) with the learning of MLCG force-field parameters based on the “force-matching” approach28. This connection, in turn, suggests novel modifications for both classes of models. The corresponding methodologies are implemented in public code bases and demonstrated by creating MLCG force-fields of the Chignolin46, Trp-Cage47, NTL948, and Ubiquitin proteins using minimal amounts of training data.

Results
Combined force matching and denoising for learning CG force-fields

Learning from forces. We train a CG neural network potential Uθ(R) with neural network parameters θ as a function of the CG coordinates, R. Force-matching minimizes the mean squared difference between the atomistic forces f(r) at configuration r, projected onto the CG space, and the forces of the CG force-field. The CG coordinates R are usually defined as a linear mapping of the atomistic coordinates: R=Mr. The effective CG force-field which produces the same equilibrium distribution as the corresponding atomistic system, referred to as the “many body potential of mean force” (UPMF), minimizes the loss function6,28:

L(θ) = 〈∥Fθ(R) − Tf(r)∥2〉r (1)

where Fθ(R)=−∇Uθ(R) is the CG force-field generated by the learned CG potential, 〈⋅〉r denotes an ensemble average over the equilibrium distribution of the atomistic system, and T transforms atomistic forces into forces on the CG particles28,36,49.

To accurately approximate the potential of mean force UPMF implied by the atomistic simulation, a flexible functional form is needed for Uθ(R), and neural networks have been shown to be a formidable choice21. However, training neural network CG force-fields for biomolecular systems requires a lot of data, e.g., many diverse configurations with force labels36. Here, we propose a much more data-efficient approach that leverages not only the force labels but also the distribution of configurations R in the training data while avoiding repeated long simulation of the CG model8,9,33,38,50,51.

Learning from noise. The proposed training modifications extend denoising score matching42, which is the central idea in DDMs43–45, a class of machine-learning algorithms trained to transform noise variates to approximate the distribution of training data. Analogous to distribution-based force-field learning8,9, DDMs learn the distribution present in a data sample without pre-recorded force information; instead, this information is learned through the addition and removal of noise.

During DDM training, data are first corrupted to different extents with additive noise, with maximal corruption giving an uninformative prior distribution and no corruption corresponding to the original data distribution. For simplicity in derivation and implementation, the noise is usually set to follow anisotropic Gaussian distribution, and the extent of corruption (i.e., the noise level) corresponds to the variance of the Gaussian noise. This corruption process is captured by introducing a conditional density κ(r0|r), which describes noising a training data sampler to obtain the noised data r0. We can interpret −logκ(r0|r) as a “noise energy" and its negative gradients as “noise forces”, Fκ=∇r0logκ(r0|r). A neural network Sθ(r0), called the “score network", is trained to predict the noise forces by minimizing a loss function. This loss function is defined as a sum of individual denoising score matching losses for each distortion level. The loss for a single distortion level can be interpreted as force matching (Eq. 1) further averaged over the noise variables:

LDSM(θ) = 〈Eκ ∥Sθ(r0) − Fκ(r0,r)∥2〉r (2)

where the brackets 〈⋅〉 denote the expectation over the data distribution via r and Eκ is the expectation over the noise variables r0. DDMs have been shown to be successful in various tasks52, including modeling CG protein conformational landscapes by fitting the training data distribution39,53.

Combining noise and forces in coarse-graining. We propose to augment Eq. (1) by adapting the denoising described in Eq. (2) for a single noise level through the definition of a modified loss function:

Lκ(θ;T) = 〈Eκ ∥Fθ(R) − Tf(R,r)∥2〉r (3)

where the force f(R,r) is now defined as a sum of the atomistic MD forces on fine-grained coordinates r and noise forces defined over an extended system with coordinates (R, r) with energies specified in thermal units (kBT=1):

f(R,r) = −∇u(r) + ∇logκ(R,r) (4)

The force map T in Eq. (3) projects forces on the extended system to the CG coordinates R.

In the case of κ(R,r), where M is the mapping used in standard CG, κ describes the probability density of obtaining a given R conditioned on r (see Supplementary Information). Similar to M, κ is a modeling choice selected before training that defines the multiscale relationship between resolutions. Although arbitrary κs can be used, in this work we focus on κs defined by probabilistically extending a given CG mapping M. Specifically, we let κ follow anisotropic Gaussian distribution centered around Mr:

κ(R,r) ∝ exp[−(Mr−R)TΣ−1(Mr−R)/2] (5)

where the covariance matrix Σ is defined to be σ2I with variance σ2 referred to as the noise level; σ is a hyperparameter that improves training on finite datasets but introduces bias into the training objective (see Supplementary Information). Practically, for given r, R can be sampled by first applying the map M and then adding scaled standard Gaussian noise. Training consists of minimizing Eq. (3) by repeatedly drawing atomistic MD simulation frames, generating random noised counterparts by conditionally sampling κ, combining saved force and generated noise information via Eq. (4), and subsequently performing gradient-based optimization to minimize the prediction error in Eq. (3) (see Supplementary Information). Although this approach has fundamental connections to DDM training, no DDM is directly used in this procedure.

In Eq. (1), the force map T projects the atomistic forces into the CG space. Different choices of T can be made for a given choice of CG coordinates36 (see Supplementary Information). In the context of Eq. (3), T also specifies how to combine distributional and atomistic force information. T may be defined to include only contributions from κ, facilitating force-field optimization when no atomistic forces are available; the corresponding T is referred to as Tnoise. In the case of Tnoise, training in the proposed framework is equivalent to the score-matching training of a DDM: Eq. (3) reduces to Eq. (2) at a fixed noise level. On the other hand, Eq. (3) reduces to Eq. (1) if a κ concentrated at Mr is considered.

Critically, however, T may instead be defined to include contributions from both saved atomistic forces and noising, allowing the seamless combination of distributional and force-based information during training. Preliminary work has shown that combining reference force information with noising may improve performance in the low-noise stage of DDMs in two dimensions54; the current work demonstrates that the theory of bottom-up CG can unify this information to create state-of-the-art MLCG force fields on complex systems with drastically reduced amounts of training data while retaining a rigorous connection to the atomistic system.

Improved MLCG force-fields. We demonstrate the performance of MLCG force fields trained to minimize Eqs. (1 or 3) for Chignolin46, Trp-Cage47, NTL9 (K12M mutant)48, and Ubiquitin; results for Chignolin are provided in the Supporting Information.

Chignolin, Trp-Cage, and NTL9 have previously required millions of frames for accurate MLCG force-field training22,55; Ubiquitin has additionally required that these training data be drawn from simulations of multiple biological systems23. Force-fields were created at a CG resolution of one site per α-carbon and trained using batched gradient-based optimization. When minimizing Eq. (1), batches were used without modification to estimate the gradient. However, in the case of Eq. (3) with a kernel defined in Eq. (5), batches were drawn and augmented with standard Gaussian noise. This procedure is described in Fig. 1. When minimizing Eq. (1), T is then defined to minimize 〈∥Tf∥2〉 as in previous work36 (see Supplementary Information). When minimizing Eq. (3), two T were used: the first combined information from both atomistic and noise-derived forces from the minimization of 〈Eκ ∥Tf∥2〉 (see Supplementary Information for details, in particular Fig. S1), and the second used only noise-related information, T=Tnoise.

Trp-Cage. MLCG models of Trp-Cage47, a 20-residue miniprotein, were optimized using training sets of varying sizes and κs of different variances (different noise levels) to investigate the data efficiency of the proposed learning procedures. Training samples were extracted from adaptively sampled short atomistic MD simulations22 at 350K that do not distribute according to the many-body potential of mean force (PMF); for model validation, this data was reweighted using a Markov State Model (MSM)56. Model accuracy was quantified using low-dimensional free energy surfaces (FESs) generated by simulating the CG force-field using unbiased MD. FESs were calculated along low-dimensional projections capturing the folding-unfolding process as revealed by TICA57 on the atomistic trajectories (Fig. 2). In Fig. 2a, b, we focus on the distribution on the first TIC dimension resolving the folding-unfolding transition, while the quantitative comparisons in Fig. 2c are made over histograms on the first two TICs. An analysis of the fraction of native contacts shows that the formation order of different structural motifs are well recovered by the models, demonstrating an application in rare-event sampling of folding-unfolding transition states (see Supplementary information IIE). We also include in the Supplementary Information the two-dimensional FESs for models and references over TIC1–TIC2, as well as RMSD–Rg (radius of gyration).

Models trained on atomistic forces without noise using Eq. (1) recover the reweighted FES (despite the biased training set) with sufficient training data (1.6M samples, Fig. 2a, left panel). In contrast, minimizing Eq. (3) using only noise information produces samples biased towards the distribution of training data (Fig. 2a, left panel). In this high-data regime, combining these sources of information by minimizing Eq. (3) with an optimized T results in similar accuracy as traditional force-matching; however, this approach greatly increases performance at lower training set sizes, yielding significantly more accuracy than traditional force-matching when fewer than 1M samples are available. Strikingly, training with only noise information is as accurate (for noise level, i.e. variance, 5 × 10−4Å2), or even more accurate (for noise level 0.003 and 0.005Å2), than conventional force-matching (Eq. 1) in this low data regime (Fig. 2a, middle; c, right panel); moreover, this data efficiency is further increased by including force information. We note that high noise levels (>0.01Å2) can be detrimental, perhaps due to breakdown of the prior energy terms in MLCG training (see Supplementary Information for details).

NTL9. The proposed training methodologies were additionally validated on NTL9 (K12M)48, a 39-residue protein. As in the case of Trp-Cage, models were optimized using training sets of varying sizes and κs of different variances (different noise levels). NTL9 represents a significantly more challenging learning target than Trp-Cage, featuring a slower folding-unfolding transition and several folding intermediates15,16,58 (Fig. 3d). Training samples were similarly extracted from adaptively sampled short atomistic MD simulations22 at 350K that do not reflect the many-body PMF; in this case, creation of a converged MSM was not feasible (see “Methods” section). Instead, millisecond-length trajectories generated at 355K by D. E. Shaw Research (DESRES) using the Anton computer with the same force-field and molecular topology were utilized as an equilibrium reference16. The distribution of samples differs significantly between the training data and the reference Anton simulations (Fig. 3a, e). We note that, as revealed by refs. 58,59, even these long Anton simulations may deviate from the underlying many-body PMF for minor states.

Similar to Trp-Cage, models trained using traditional force matching and a combination of force and noise information accurately approximate the distribution of the reference data despite the biased training data; in comparison, those trained using only noise information are biased toward the training distribution (Fig. 3a). We note that models trained using only noise information still exhibit a bias towards the correct equilibrium distribution, while in principle should reproduce the distribution of the training data. This bias maybe induced by the CG SchNet20 architecture (see the “Discussion” section). At lower training sizes, the combination of force and noise information greatly increases accuracy. Interestingly, at intermediate noise levels, the combined model can produce a reliable MLCG force-field with only 1% of the whole training set (76k frames, bottom left of Fig. 3c). We note that the PMF error saturates at 7.5kBT for the other models (noise only or forces only) at lowest training set sizes, as this corresponds to a model completely destabilizing the folded state and folding intermediates (Fig. 3e). At an intermediate noise level, the model is also capable of reproducing the majority of the native contact trends (see Supplementary Information).

Ubiquitin. To demonstrate the applicability of the proposed training procedure to larger biomolecular systems, Ubiquitin (PDB: 1D3Z, 76 residues) was also modeled. Simulating the equilibrium folding/unfolding process of Ubiquitin at 300K has remained a long standing challenge in the simulation community60,61 due to the high stability of the folded state. For example, even on the Anton specialized supercomputer, an elevated temperature (395K) was required61 to observe folding and unfolding transitions, with total simulation lengths reaching more than 7 ms.

Here, we employ the proposed approach to simulate equilibrium folding/unfolding of Ubiquitin with a Cα MLCG trained on short out-of-equilibrium atomistic simulations. Training data were obtained from atomistic MD performed at 300K using the AMBER99SB-ILDN force field62; these simulations were performed using two setups. First, short unbiased simulations initialized from the crystal structure were used to characterize the fluctuations present in the folded basin. Second, constant-velocity steered MD was used to unfold the protein by applying a stretching force to both termini; this procedure forced the simulation to sample conformations where the hydrophobic core of the protein was exposed. These steered MD simulations were performed using an aggressive velocity rate (0.69 nm/ns) and cannot be reweighted to obtain the equilibrium distribution of the system; as a result, we do not have a reference free energy profile from atomistic simulations to which to compare our CG model. The accelerated pulling rate allowed the unfolding of the protein with steered MD simulations in 230 ns, which could be accomplished in less than 3 weeks using a single Nvidia-RTX3090 GPU. The total size of the training dataset, consisting of short folded simulations and steered MD simulations, contained 250,000 instances of configurations and forces, distributed as illustrated in Fig. 4a (leftmost panel). Additional details are provided in the “Methods” section.

Several MLCG models for Ubiquitin were trained on this out-of-equilibrium atomistic dataset, and for each model, equilibrium FESs were estimated using parallel tempering (see Supplementary Information IIB). Figure 4 shows that the folded state is not sampled using the force-only MLCG model—it is fully unfolded. In contrast, MLCG models trained on either atomistic and denoising or denoising-only forces both possess metastable folded and unfolded states. However, only the model trained with a combination of atomistic and denoising forces yields the folded state as the most stable state, as shown in Fig. 4b. It is interesting to note that although both atomistic and denoising and denoising-only models exhibit high folding barriers, neither model visits the highly distorted extended states that were generated by steered MD and are present in the training data.

The only existing folding/unfolding atomistic simulation of Ubiquitin is the aforementioned one performed by DESRES on the Anton supercomputer at 395K, and with a force-field (CHARMM22*63) different from ours. The behavior of our trained MLCG models can thus be compared only with the Anton simulation on a qualitative basis. We also note that the DESRES simulations visualized in Fig. 4b (lower panel) were performed with a periodic box length of 5.8 nm; as a result, it is unclear whether the unfolded basin is affected by self-interaction artifacts. In contrast, the steered MD simulations used to generate the training data employed a box length of 15 nm (see the “Methods” section).

Methods
Optimizing CG force-fields
MLCG force fields were optimized to minimize their force prediction error on CG conformations. The CG resolution was defined by the removal of all atoms except the alpha carbons of each amino acid; this is algebraically encoded in M or κ, depending on the training equation. CG SchNet20 (a modified SchNet75 graph neural network plus prior energy terms) with two interaction blocks was used to represent all CG force-fields20. Auto-differentiation was used to obtain the forces on each CG bead.

When optimizing force-fields using traditional force matching, established procedures were used19,20,36. Positions and forces were drawn from atomistic MD simulations and mapped to the CG resolution using preselected M and T matrices and used to generate batched gradient updates using the ADAM optimizer.

When minimizing Eq. (3), the gradient update was modified as follows. Batches were first drawn and mapped to the non-noised alpha carbon resolution. Second, 0-mean Gaussian noise with diagonal covariance was added to the positions; collectively, this corresponds to κ(R,r) defined to be a Gaussian centered at Mr. Third, forces were modified according to Eq. (4) with a preselected T. T was defined either to minimize 〈Eκ ∥Tf∥2〉 or, in the case of noise-only forces, directly defined via Eq. (S4) (see Supplementary Information).

To investigate the data efficiency of different training strategies and noise levels, the training and validation set size was varied. Smaller datasets were created from the full dataset via striding, ensuring that the conformational distribution did not change. For each training condition, a representative model was selected for MD analysis using the following criteria. After the initial decrease in the validation loss, if a consistent increase was observed over multiple epochs, the model at the lowest reached validation loss was selected. In case of no obvious increase (e.g., for large training sizes with denoising information), training was stopped at a certain number of epochs (e.g., 200 for NTL9 models on full training data) and used for simulation analyses. Note that there were cases where the model quality seemed to be sensitive to stochastic fluctuations during training or deteriorated over epochs despite validation losses continuing to gradually decrease, especially for NTL9 models at intermediate noise levels (e.g., for a level of 0.003 Å2). If simulation instability was observed, an earlier epoch was selected. More details can be found in the Supplementary Information.

Optimizing T
All force maps were optimized using the aggforce package76 (version 1.0.1); force maps including noise were generated using the stagedjoptgauss_map and stagedjslicegauss_map methods. Supporting libraries include numpy77 (version 1.26.4), jax78 (version 0.4.35), pandas79 (version 1.15.2), scipy80 (version 1.1), and OSQP81,82 (version 0.6.7.post3). Force maps were optimized using an L2 regularization of 103 for atomistic contributions and 5 for post-map noise contributions (see Supplementary Information); forces were considered in units of kcal/(mol⋅Å). Force map optimization was performed using 37, 302, 350,000, 500,000, and 784,000 MD frames for Chignolin, Trp-Cage, NTL9, and Ubiquitin, respectively.

Reference atomistic MD simulations
MLCG models were trained and validated against solvated all-atom MD simulations. For training Chignolin, TRPcage, and NTL9 models, a dataset generated on GPUGRID83 was used22, which utilized thousands of short simulations seeded with an adaptive sampling strategy. Collectively, this dataset provides approximately 1.8M frames of coordinates and forces from 3744 simulations (~50 ns each) for Chignolin, 2M frames from 3940 simulations (50 ns each) for Trp-Cage, and 9.5M frames from 47,599 trajectories (20 ns each) for NTL9. The simulation procedure and data availability are described in ref. 22. Owing to the long timescale of major conformational transitions, the data distribution deviates from the true Boltzmann distribution for the all-atom systems. For Trp-Cage, we obtain the reference FES with MSM reweighting as in ref. 36. For NTL9, due to difficulties in obtaining implied timescales consistent with Markovian dynamics56, data generated using the Anton supercomputer by D. E. Shaw Research was used as a reference for the equilibrium distribution. This data comprises four long simulations (2.9 ms in aggregate time) with the same force-field and solvation setup16. The Anton simulations were performed at a temperature of 355 K instead of 350 K of the training data from GPUGRID, implying that the folded state is expected to be slightly destabilized relative to the targeted many-body potential of mean force.

For ubiquitin, a dataset was generated using the AMBER ff99SB-ILDN force field62 at 300K. To sample fluctuations around the folded state, 28 independent trajectories were generated, all of them starting from the reference crystal structure of the native state, for a total of 158 μs of simulation time; these trajectories were qualitatively identical. To obtain conformations further away from the folded basin, a single trajectory was generated using steered MD by applying a force constant of ~120 kcal/(mol nm2) to the alpha carbons of the first and last residues of the protein. This pulling force was used to stretch the protein from the end-to-end distance of 3 nm to 8.15 nm. The pulling force was applied as a harmonic restraint over the end-to-end distance for 115 uniformly spaced partitions of the mentioned interval. For each of these windows, 4 ns of simulation were run with data recorded for only the last 2 ns. In total, the steered MD dataset amounts to 230 ns of simulation time. We note that the DESRES simulations of Ubiquitin63 cannot be used for training as they do not contain forces.

For the atomistic Ubiquitin simulations, a non-bonded cutoff of 9 Å and switch distance of 7.5 Å were applied. Long-range electrostatics were handled using the Particle Mesh Ewald method, and H-bonds were constrained. The system was constructed using a cubic box and solvated using TIP3P water and Na+, Cl− ions for a salt concentration of 0.1 μM. For unbiased folded simulations, the side of the box was set to 6.5 nm, creating a system with 26,784 atoms. For the steered MD simulations, the size was increased to 15 nm to account for the full unfolding of the protein, which produced a system with 325,004 atoms. After preparing the solvated boxes, the energy of the system was minimized, and then 8 ns of NPT equilibration was performed using Monte Carlo Barostat and Langevin dynamics with a friction coefficient of 0.1 ps−1 and a 2 fs timestep. After equilibration, production simulations were performed using a Langevin dynamics with a friction coefficient of 0.5 ps−1. The preparation of the systems was performed with GROMACS84 (version 2022.5) while the equilibration and production simulations were performed in OpenMM85 (version 8.0.0).

CG MD simulations
Trained MLCG models of Chignolin, TRPcage, and NTL9 were evaluated by performing 100 MD simulations at the same temperature as the training data (350 K). The starting structures for simulation were randomly sampled from the training set. The simulations used the Langevin thermostat with a friction coefficient of 1 ps−1 and a timestep of 2 fs. For Chignolin, Trp-Cage, and NTL9, each trajectory was simulated for 2M, 5M, and at least 10M time steps, respectively. The simulation time was sufficiently long to allow for several transitions along the first TIC. Note that a small number of NTL9 models exhibited integration instability; this was addressed by re-initialization of the corresponding simulation. This might be attributed to the imperfect functional form and fitting of prior terms based on Gaussian-noised statistics. For details of the implementation, we refer to ref. 36 and Supplementary Information.

Ubiquitin models were evaluated using both Langevin dynamics and PT simulations; the Langevin simulations were unable to converge the corresponding FES within 20M time steps due to a high transition barrier. Using a set of 10 geometrically-spaced temperatures between 284.1 and 362 K and an exchange interval of 4 ps, PT simulations were initialized from 20 different structures, including configurations from the folded basin and the transition region (Fig. S5). The observed exchange acceptance rate for all temperatures was higher than 0.386. PT Simulations ran for at least 13M time steps using a Langevin thermostat with a friction coefficient of 1 ps−1 and a timestep of 4 fs.

Data availability
The all-atom MD simulations used for the model training for Chignolin, TRPcage, and NTL9 are the same as in ref. 22 and are publicly available at https://zenodo.org/records/815534387. In-house simulations for Ubiquitin and all the trained models presented in this study are publicly available at https://zenodo.org/records/1875663888. Reference DESRES simulations from existing publications16,61 were requested from D. E. Shaw Research and served only for comparison and are not essential for model training. Source data are provided with this manuscript. Source data are provided with this paper.

Code availability
Noise-based data transformations were implemented in the aggforce package (version 1.0.1), which is publicly available at https://zenodo.org/records/1878468576, and training was performed using the mlcg package (version noise-alpha), which is publicly available at https://zenodo.org/records/1877035189. A demo showing how to run the code is publicly available at https://zenodo.org/records/1875663888.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This article introduced a machine-learning-based coarse-grained force field training approach, which improves the MLCG\ntraining by combining force-based and noise-based learning. This training method is an interesting approach, but in my\nreview of related works of literature, I found a similar work in arXiv preprints first submitted in May 2023: arXiv:2305.01243v1,\nwhich proposes a method called cycle coarse-graining (CCG). This preprint article claimed that CCG can perform\nboth coarse-graining (CG) and fine-grained reconstruction (FCR) operations. Both approaches use generative learning, but their\nspecific implementations are different. The noise-based generative learning proposed in this paper is more accessible to\nimplement than the generative adversarial network (GAN) on which the CCG is based. However, the CCG method can\nperform FCR in addition to CG. As a different CG approach, the method proposed in this article still has value for publication.\nHowever, since a similar approach has already been posted, this article should be majorly revised to compare it with the\nCCG method systematically.\n1. The authors should systematically compare their approach to generative learning with the CCG. In particular, the loss\nfunction of the method should be compared with that of the CCG (Equation S7 in arXiv:2305.01243v1) to see if there is a\nfundamental conceptual difference between the two. In addition, the authors could also discuss the advantages and\ndisadvantages of the two approaches in generating learning.\n2. Is it possible for the method to do FCR in some way like the CCG?\n3. The method is claimed to be a “data efficient” method, and the authors of the CCG also claim that their method can be\ntrained with fewer samples. Can the authors compare the difference in training samples between the two methods? Who can\nuse less data to train a CG force field with the same system and the same accuracy?\n4. The arXiv:2305.01243v1 article trained the CG force field of the peptide Chignolin. Could the authors also train the force\nfield of the same system for comparison? In addition, when demonstrating the effect of a peptide CG force field, it would\nmake more sense to project the free energy on some physically meaningful reaction coordinates, such as Rg and RMSD, as\nthe CCG article does, rather than on the TICA coordinates. FES of the already trained Trp-cage and NTL9 force fields\nshould also be drawn this way.\n5. The CCG approach also claimed to solve the rare event sampling problem through active learning. They trained a CG\nforce field for chemical reactions in addition to the protein force field. Can the authors indicate whether the methodology of\nthis paper can be easily trained for such systems involving rare events?\nIn summary, this paper is methodologically innovative as a new generative learning-based approach to coarse-grained force\nfield training. This article is worthy of publication, but before doing so, it should be systematically compared with already\nreleased methods.", "reviewer_2": "This manuscript studies the effect of adding a noise bias to the widely force-matching objective function for bottom-up\ncoarse-grained modeling. The authors explore numerically on two small peptides how the quality of the many-body potential\nof mean force (mb-PMF) is represented with force mapping procedures that are deterministic or noisy with a particular focus\non the size of the data set used to train the coarse-grained model. They draw analogies to denoising diffusion models but I\ndo not think there is yet a strong theoretical justification for this empirically powerful approach.\nOverall, I think this is a nice contribution to the large literature on building coarse-grained models and would be happy to see\nit published in its current form. The results are quite compelling: the numerical results demonstrate unambiguously that the\nrepresentations of the mb-PMF improve with appropriate noising of the training data and furthermore, the noising procedure\nmassively reduces the total amount of data required to build a CG forcefield. Despite the fact that the configuration space is\nwell-sampled on these data rich problems, the conventional force matching objective does not perform anywhere near as\nwell as the noised objectives.\nI have a couple suggestions that I believe could improve the presentation of the paper, which I would like the authors to\nconsider:\n- How the loss in (3) is evaluated is not well-described in the text. It would be beneficial to explain the ancestral sampling\nprocedure shortly after equations (3) and (4) because it's an important aspect of actually using this objective. In particular,\nthe procedure for evaluating (4) of $f(r,R)$ is not clear in the main text, though well explained in the appendix.\n- The fact that the distribution $\\kappa$ a modeling choice with hyperparameters also merits a clear discussion. I feel that\nprevious work from the authors on optimal aggregation of forces could be better leveraged to explain the additional noise as\naccounting for uncertainty in the projected forces.\n- In Fig. 2, the caption mentions that a single \"noise level\" is visualized and that term appears throughout the text\ninterchangeably with \"noise variance\". While I think it's clear from the units what the quantity means, it should be defined\nand explained somewhere.\n- For the coarse-grained simulations, it is mentioned that 100 trajectories are collected, but where the initial conditions are\nobtained is not explained. Do these also have good coverage of the mb-PMF?", "reviewer_3": "In this paper from Noe and Clementi labs propose a hybrid approach that combines denoising score matching (used in\ndiffusion models) and multiscale coarse graining (force matching) to improve training efficiency in CG force fields while\nreducing the need for large datasets. This method has been tested on proteins such as Trp-Cage and NTL9, demonstrating\na significant reduction in data requirements by a factor of 100 while maintaining accurate simulation results. Their model can\ngenerate accurate free-energy surfaces and simulate the folding-unfolding process of proteins even with minimal training\ndata. The authors tested the performance with different noise levels and varying amounts of training data, showing that\nincluding noise information improves accuracy, especially when data is scarce. The paper is well written and applying the\ndiffusion plus noise augmentation is the innovative parts. I think also the paper might be more suitable for specialized\njournals such as JCIM or JCTC. Here are the comments and concerns I have:\n1. Obviously, the biggest concern is the generalizability and the ad-hocness of the method to the protein of interest. While\nthe method works well for the tested proteins, its scalability to even more complex biomolecules (or different types of\nmolecular systems) remain to be fully explored. I strongly suggest pre-training on multiple small proteins and test on a\ncompletely hold out protein to see the real effect of denoise pretraining. I mean for example\npre-train on TRP-cage and NTL9 and test on the Ubiquitin or a small protein. You can fine-tune with a few first step of trajectory\n(and that can be an ablation study).\n2. Authors should explain and clarify the inference time for DDM. Diffusion models are notoriously slow on inference\nspecially for high quality generation and when number of denoising steps go up. A study to show this time is not adding\nmore compute time as opposed to the reduction of the compute time authors pitched for the paper.\n3. The reliance on pre-trained atomistic simulations, although reduced, is still a limitation in terms of accessibility for users\nwithout high-performance computational resources. For general proteins and specially larger ones such as receptors, such\ntraining data ( as you obtained from DESRES) might not be available. Again, as I mentioned in the first comment, I would\nlove to see the transferability to larger proteins and different proteins.\n4. I have a suggestion, which might be a bit out of scope but makes the work stronger: How about other pretraining\ntechniques such as mask pretraining or bond/atom deletion augmentation? It would be interesting to explore those\npretraining strategies.\n5. In equation 4 and for total loss, I would like to see a regularization factor (Lambda) to see the effect of MD force versus\nnoise.\n6. One minor comment: some of the figures in appendix were not compiled properly and are not showing up"}, "author_response": "Response to reviewer 1\nThisarticleintroducedamachine-learning-basedcoarse-grainedforcefieldtrainingapproach,which\nimproves the MLCG training by combining force-based and noise-based learning. This training\nmethod is an interesting approach, but in my review of related works of literature, I found a\nsimilar work in arXiv preprints first submitted in May 2023: arXiv:2305.01243v1, which proposes\na method called cycle coarse-graining (CCG). This preprint article claimed that CCG can perform\nboth coarse-graining (CG) and fine-grained reconstruction (FCR) operations. Both approaches use\ngenerative learning, but their specific implementations are different. The noise-based generative\nlearning proposed in this paper is more accessible to implement than the generative adversarial\nnetwork (GAN) on which the CCG is based. However, the CCG method can perform FCR in\naddition to CG. As a different CG approach, the method proposed in this article still has value\nfor publication. However, since a similar approach has already been posted, this article should be\nmajorly revised to compare it with the CCG method systematically.\nWe appreciate the reviewer’s kind comments about our work. CCG (arxiv:2305.01243v2) is\na novel method for multiscale learning that utilizes normalizing flows trained to link between a\nfine-grained reference model and a coarser resolution to simultaneously learn how to reconstruct\nfine-grained samples and approximate the free energy surface at the coarser resolution. This con-\nnection allows for multiple new training approaches to be defined. The approach proposed in our\nmanuscript does not directly use any generative machine learning architecture such as normalizing\nflows, generative adversarial networks, or diffusion models; instead, it extends ideas from denoising\nscore matching (a technique central to diffusion models) to train a neural network force-field at the\ncoarse-resolution. As we do not adopt any generative neural architecture that bridges resolution,\nwe cannot directly compare to many of the results in arxiv:2305.01243v2. We instead here provide\ndiscussions on different aspects of the draft such as training residuals and applications.\nReviewer Point P1.1 — The authors should systematically compare their approach to gener-\native learning with the CCG. In particular, the loss function of the method should be compared\nwith that of the CCG (Equation S7 in arXiv:2305.01243v1) to see if there is a fundamental con-\nceptual difference between the two. In addition, the authors could also discuss the advantages and\ndisadvantages of the two approaches in generating learning.\nAuthor reply: Eq. S7 in arXiv:2305.01243v1 effectively describes a linear combination of 3 loss\nterms (see S3-S5). The first two terms correspond to losses guiding the reconstruction of atomistic\nconfigurations, which we do not perform. The third term corresponds to traditional force matching\nwithout the addition of noise. In our manusript we discuss extensively the connections between our\nproposed method and traditional force-matching, which is the common term between our loss and\nthe CCG loss. We can not directly compare with the other two terms in the CCG loss, as we do\nnot perform atomistic reconstruction, but we have now noted the utility of our method in the general\ncontext of backmapping in the discussion and provided reference to the CCG article (lines 333-347).\nOther equations presented in arXiv:2305.01243v1 (e.g., S3, a different component of S7) typically\nfocus on methods performing backmapping and often require that configurations at the resolution of\nthe reference energy function be reconstructed. While such terms are difficult for us to use without\nsubstantially expanding our method to incorporate backmapping, we also note that the simulations we\nstudyasourfine-grainedreferencesystemsareattheresolutionoffullysolvatedatomisticconfigurations\n1\nand we are not aware of any backmapping procedure that is capable of effectively constructing a fully\nsolvated atomistic protein structure.\nReviewer Point P1.2 — Is it possible for the method to do FCR in some way like the CCG?\nAuthor reply: Incorporating fine-grained reconstruction (i.e., backmapping) to our proposed method\nwould be a highly substantial addition to the corresponding manuscript, as the current work does not\nconsider the task of backmapping. While CCG focuses on reconstruction and relates this back to the\nvalue of the manybody PMF, our proposed method directly approximates the manybody PMF without\nreference to any reconstruction. However, we recognize that fine-grained reconstruction is an important\ntopic and we have added a section in the discussion on how backmapping may be affected by the noise\naddition discussed in the manuscript (lines 333-346).\nReviewer Point P1.3 — The method is claimed to be a “data efficient” method, and the\nauthors of the CCG also claim that their method can be trained with fewer samples. Can the\nauthors compare the difference in training samples between the two methods? Who can use less\ndata to train a CG force field with the same system and the same accuracy?\nAuthor reply: WehavenowaddedChignolintooursetoftestproteins,asintheCCGarticle. However,\nwe cannot use the same training data as the CCG work as these data do not contain forces on which to\ntrain. Conversely, as the code of CCG is not available, we cannot rerun their code on our dataset for\ncomparison.\nWe further note that, while the current manuscript focuses on reproducing molecular distributions\npresent in the training data, the ultimate utility of the proposed method lies in training models of\nthis type on larger training data distributions in hopes of extrapolating to novel proteins not in the\ntraining set. We have demonstrated this in a recent manuscript (Nature Chemistry 17, 1284–1292\n(2025)). While the process of estimating the density (e.g., histogramming) of an existing molecular\nsimulation along a low dimensional CV and subsequently combining low-dimensional sampling methods\nwith backmapping can be viewed as a generative model of a given simulation, it is currently unclear\nhow this procedure can be adapted to extrapolate to proteins outside the training set.\nReviewer Point P1.4 — The arXiv:2305.01243v1 article trained the CG force field of the\npeptide Chignolin. Could the authors also train the force field of the same system for comparison?\nIn addition, when demonstrating the effect of a peptide CG force field, it would make more sense\nto project the free energy on some physically meaningful reaction coordinates, such as Rg and\nRMSD, as the CCG article does, rather than on the TICA coordinates. FES of the already trained\nTrp-cage and NTL9 force fields should also be drawn this way.\nAuthor reply: Thank you for the suggestion. We have now added results for Chignolin in the SI\n(Supplementary Information, Lines 301-319) and now provide RG/RMSD FEs in the SI (Supplementary\nInformation, Figures S7, S9 and S12).\nReviewer Point P1.5 — The CCG approach also claimed to solve the rare event sampling\nproblem through active learning. They trained a CG force field for chemical reactions in addition\nto the protein force field. Can the authors indicate whether the methodology of this paper can be\neasily trained for such systems involving rare events?\ncontain rare events. For example, they can be trained on data from smaller, easier to sample reference\nsystems and then used to characterize targets not present in the training set (e.g., our recent Nature\nChemistry 7, 1284–1292 (2025) article). Alternatively, force-matched models can be combined with\ndata generated by enhanced sampling and subsequently used to probe rare events. We have now added\nanexampleofthelatterinthemanuscript,byusingthelearnedCGmodeltostudythefolding/unfolding\ntransitionofUbiquitin, whichisimpossibletosamplebyatomisticsimulationsat300K(evenD.E.Shaw\nResearch had to use elevated temperature to simulate it on the Anton computer). Directly utilizing\nworkflows such as that given in arXiv:2305.01243v2, page 11, Fig 4, is only feasible when reconstruction\ncreates samples at the resolution of the reference system. As noted previously, to our knowledge, no\nsuch method exists for mapping between coarse-grained and all-atom solvated protein structures, which\nwould correspond to substantial addition to the manuscript.\nResponse to reviewer 2\nThis manuscript studies the effect of adding a noise bias to the widely force-matching objective\nfunction for bottom-up coarse-grained modeling. The authors explore numerically on two small\npeptides how the quality of the many-body potential of mean force (mb-PMF) is represented with\nforce mapping procedures that are deterministic or noisy with a particular focus on the size of\nthe data set used to train the coarse-grained model. They draw analogies to denoising diffusion models but I\ndo not think there is yet a strong theoretical justification for this empirically powerful approach.\nOverall, I think this is a nice contribution to the large literature on building coarse-grained\nmodelsandwouldbehappytoseeitpublishedinitscurrentform. Theresultsarequitecompelling:\nthenumerical resultsdemonstrateunambiguouslythat therepresentationsof the mb-PMFimprove\nwith appropriate noising of the training data and furthermore, the noising procedure massively\nreduces the total amount of data required to build a CG forcefield. Despite the fact that the\nconfiguration space is well-sampled on these data rich problems, the conventional force matching\nobjective does not perform anywhere near as well as the noised objectives.\nI have a couple suggestions that I believe could improve the presentation of the paper, which I\nwould like the authors to consider:\nReviewer Point P2.1 — How the loss in (3) is evaluated is not well-described in the text. It\nwould be beneficial to explain the ancestral sampling procedure shortly after equations (3) and (4)\nbecause it’s an important aspect of actually using this objective. In particular, the procedure for\nevaluating (4) of f(r,R) is not clear in the main text, though well explained in the appendix.\nAuthor reply: Thank you for pointing this out. We moved the most relevant definitions from the\nSupplementary Information to the main text (lines 121-145), and we hope this point is more clear now.\nReviewer Point P2.2 — The fact that the distribution κ a modeling choice with hyperpa-\nrameters also merits a clear discussion. I feel that previous work from the authors on optimal\naggregation of forces could be better leveraged to explain the additional noise as accounting for\nuncertainty in the projected forces.\nincluded in the Discussion section (lines 307-319), whereas the theoretical reasoning appears in Supple-\nmentary Information (lines 85-103). Briefly, the introduction of the noise kernel κ enables\n1. Combining information across nearby configurations.\n2. Synthesizingtheinformationfromthedenoisingscoresandtheatomisticforces, whichhaverather\nindependent sources of uncertainty.\nBotheffectscontributetothereductionofuncertaintyinthefinalCGforces. However,sincethetraining\ndata distribution (and thus the information from noise forces) is often biased, suitable κ need to be\npicked like hyperparameter tuning to maximize the benefit from this combination and yet to avoid the\ndiscrepancies from the biases.\nReviewer Point P2.3 — In Fig. 2, the caption mentions that a single ”noise level” is visualized\nand that term appears throughout the text interchangeably with ”noise variance”. While I think\nit’s clear from the units what the quantity means, it should be defined and explained somewhere.\nAuthor reply: We thank the reviewer for kindly pointing this out. We have included a clear definition\nof the noise level / extent of noise in Section I(b) (lines 108-110) and mentioned it again in Section I(c)\n(lines 136-138).\nReviewer Point P2.4 — Forthecoarse-grainedsimulations,itismentionedthat100trajectories\nare collected, but where the initial conditions are obtained is not explained. Do these also have\ngood coverage of the mb-PMF?\nAuthor reply: The starting structures were randomly picked from the training set. We have included\nthis missing description to the Methods section (lines 443-455, 456-458). The sampled structures cover\nthe mb-PMF reasonably well, since most samples are picked from the populated metastable states.\nAlthough this might bias the performance of all models towards recovering the mb-PMF, we observed\nthat inaccurate models tend to lead the systems away from the correct states in a small number of time\nsteps, yielding very different probability distributions from the reference.\nResponse to reviewer 3\nIn this paper from Noe and Clementi labs propose a hybrid approach that combines denoising score\nmatching (used in diffusion models) and multiscale coarse graining (force matching) to improve\ntraining efficiency in CG force fields while reducing the need for large datasets. This method has\nbeen tested on proteins such as Trp-Cage and NTL9, demonstrating a significant reduction in data\nrequirements by a factor of 100 while maintaining accurate simulation results. Their model can\ngenerate accurate free-energy surfaces and simulate the folding-unfolding process of proteins even\nwith minimal training data. The authors tested the performance with different noise levels and\nvarying amounts of training data, showing that including noise information improves accuracy,\nespecially when data is scarce. The paper is well written and applying the diffusion plus noise\naugmentation is the innovative parts. I think also the paper might be more suitable for specialized\njournals such as JCIM or JCTC. Here are the comments and concerns I have:\nhocness of the method to the protein of interest. While the method works well for the tested\nproteins, itsscalabilitytoevenmorecomplexbiomolecules(ordifferenttypesofmolecularsystems)\nremain to be fully explored. I strongly suggest pre-training on multiple small proteins and test on\na completely hold out protein to see the real effect of denoise pretraining. I mean for example\npre-train on TRP-cage and NTL9 and test on the Ubiquitin or a small protein. You can fine-tune\nwith a few first step of trajectory (and that can be an ablation study).\nAuthor reply: Thank you for the suggestions. We here divide the stated concerns into two separate\ntopicsrelatedtothecurrentstudy: scalabilitytomorecomplexsystemsandtransferability, e.g., learning\nfrom small systems and predicting on larger systems. In the revised manuscript, we have now provided\nadditional experiments for the first topic by directly modeling Ubiquitin using training data drawn from\nhighly unconverged steered MD simulations (lines 237-278). Unlike the previous training datasets, these\ndata were generated using hardware resources available to a typical research group (approximately 3\nweeks of simulation using a single RTX3090). While traditional force-matched models cannot stabilize\nthe protein in its native state, the models trained with our proposed method can not only stabilize the\ncorrect metastable states, but can also approximate the unfolding order of key structure elements. We\nbelieve that this showcases the scalability of the proposed method to larger proteins using resources\navailable to a typical research group.\nOnthesecondtopic, wehavenowprovidedamoredetaileddiscussioninthemanuscriptontransfer-\nability and we summarize it here. Transferable learning is a practical way to create useful coarse-grained\nmodels, and the current authors have provided foundational workon how to dosousing traditional force\nmatching; this work was recently published in Nature Chemistry (Nat. Chem. 17, 1284–1292 (2025)).\nAs shown in that manuscript, the creation of a transferable model presents multiple substantial engi-\nneeringchallengesbeyondthetrainingobjectiveused. Expandingthiscurrentarticletoencompassthese\nchallenges would represent a quantity of work beyond what is typical for a Communications article and\nwould dilute its primary content explaining the reasoning behind a novel training objective; we note that\nthe provided numerical examples still represent state-of-the-art performance on the selected molecular\nsystems. We further note that the now-included modeling of Ubiquitin represents an application that is\naccessible to a typical research group and shows the potential of the proposed approach to go beyond\nthedatausedfortraining. Weexpectthatultimatelytheprovidedresultswillmotivatetheincorporation\nof the proposed training objectives into future transferable modeling attempts.\nReviewer Point P3.2 — Authors should explain and clarify the inference time for DDM.\nDiffusion models are notoriously slow on inference specially for high quality generation and when\nnumber of denoising steps go up. A study to show this time is not adding more compute time as\nopposed to the reduction of the compute time authors pitched for the paper.\nAuthor reply: There seems to be a misunderstanding on this point and we apologize with reviewer\nfor not clearly explaining in the original manuscript that no DDM (denoising diffusion model) was used\nin any experiments in this work. We mentioned DDMs in the manuscript as it is a well-known example\nof utilizing forces (i.e. scores) derived from a noising process, the same source of information on which\nour proposed method is built upon. Nevertheless, the methods are vastly different: A DDM aims\nto reconstruct the data distribution from pure noise-base information by training over many (usually\nover hundreds of) noise levels, while the proposed models in this work focus on the distribution and\nthe corresponding many-body PMF on a single predetermined noise level. In addition, the sampling\nprocesses are distinct: DDMs proceed step-by-step from the highest noise level downward, while our\n5\nmodels are CG potentials and can be used for molecular dynamics (i.e., they do not incur any additional\nsimulation cost relative to a traditional force-matched model). In fact, the computing time required for\ntraining is usually shorter for our proposed method than for pure-force-matching MLCG models, as our\nmodel reaches high accuracy using a smaller amount of training data. We have now included an explicit\ndiscussion on the difference between our approach and a DDM in the manuscript to clarify this matter\n(lines 144-145).\nReviewer Point P3.3 — The reliance on pre-trained atomistic simulations, although reduced,\nis still a limitation in terms of accessibility for users without high-performance computational\nresources. For general proteins and specially larger ones such as receptors, such training data (as\nyou obtained from DESRES) might not be available. Again, as I mentioned in the first comment,\nI would love to see the transferability to larger proteins and different proteins.\nAuthor reply: We appreciate the reviewer’s consensus with us on the merit of learning a transferable\ncoarse-grained model applicable to different proteins. We have already discussed this topic and its\nconnection with this work in our response to the first point above. Here, we would like to additionally\npoint out that many challenges remain to be overcome before we can reach this “holy grail”, and this\nwork addresses the data availability issue by both improving training data efficiency while maintain-\ning compatibility with learning on conformations from unconverged enhanced sampling methods (as\ndemonstrated by our now-provided application to protein Ubiquitin). As the reviewer points out, this\nsecond property is critical for learning on systems of biological relevance using data acquirable by typical\nresearch groups, and the current authors are not aware of any other biomolecular MLCG force-fields\nthat have been trained on this type of data.\nReviewer Point P3.4 — I have a suggestion, which might be a bit out of scope but makes the\nwork stronger: How about other pretraining techniques such as mask pretraining or bond/atom\ndeletion augmentation? It would be interesting to explore those pretraining strategies.\nAuthor reply: We appreciate these suggestions. Pretraining strategies such as mask pretraining and\nbond/atom deletion augmentations have been reported for models created for predicting properties of\nmolecules as well as those generating molecules directly from scratch for given tasks. Masking is also\nvital for training transformer models (e.g., large language models). However, the mentioned pretraining\napproaches are not directly relevant to MLCG models with CGSchNet, where every CG bead needs to\nbe present in order to define and compute the multibody potential of mean force at a given resolution.\nDue to the multibody nature of the prediction target, it is not immediately obvious to us what labels\n(e.g., energy/forces) we should expect the model to predict in the pretraining stage. In other words,\nthe current setup does not allow us to remove some part of the molecule for augmentations.\nHowever, duringournewapplicationtoUbiquitin, wedevelopedaregularizationstrategyfortraining\nMLCG models. In brief, for each mini batch of training, we apply the following to a small proportion\nof data points: an isotropic Gaussian noise (with a much higher variance than the level used for noise\nforces) is generated and added to the coordinates, and the force label to match is set to zero. This\nregularization discourages the neural network from predicting high-magnitude forces outside of the\nknown data distribution, which stabilizes the models during long CG simulations by preventing sudden\nblow-ups due to very wrong force predictions. We have now noted this procedure in the Supplementary\nInformation (Supplementary Information lines 194-197).\nfactor (Lambda) to see the effect of MD force versus noise.\nAuthor reply: There is indeed a balance to strike between the information extracted from MD forces\nand that from noise; this balance produces a trade-off between information derived from the true\nunderlying Boltzmann distribution and that from the observed distribution of training samples. There\nare two hyperparameters to set that affect this balance: one is the variance of the noise (noise level)\nembedded in the definition of κ and the other (the “Lambda”) is implicitly included in the extended\nforce map T in equation 3. For each noise level, the corresponding regularization (Lambda) is set by\nusing an optimized form of T, to minimize the force-matching noise as we have previously proposed (see\nJ. Chem. Phys. Lett., 14:17, 3970-3979 (2023)). Essentially, we bind these two choices together: The\nselection of the noise level, along with the given training dataset, determines T. In the manuscript, we\nhave explored the effect of modifying the noise level, and have additionally compared this to the case\nof a non-optimized noise-only T (the noise-only cases) and the force-only case. As a result, we believe\nthat we have explored the trade-offs observed by changing the balance of noise and force information.\nIn summary, as the balance of noise and force information is already implicit in the T projection\nin equation (3), there should be no additional factor in front of either term in Eq. 4 to represent\nthis balance. To avoid this confusion, we have reworded the derivations. We also included pointers\nto the Supplementary Information and Fig. S1 to help clarify the theoretical meaning of these terms\n(Supplementary Information lines 222-232).\nReviewer Point P3.6 — One minor comment: some of the figures in appendix were not\ncompiled properly and are not showing up\nAuthor reply: Thank you for pointing this out. It should be fixed now.\n7"}, {"round": 2, "reviewer_comments": {"reviewer_1": "In the revised manuscript, the authors’ modifications and discussions of relevant details are commendable. The newly\nincorporated Chignolin system and free energy surface analysis based on Rg/RMSD enhance the comparability of results.\nSupplementary Figures S7, S9, and S12 in the SI indeed clarify the physical significance.\nHowever, in addressing the core issue I mentioned, that is, the comparison with the CCG method (arXiv:2305.01243v1).\nThere are clear shortcomings, suggesting an attempt to sidestep the crucial points. I explicitly requested a comparison of the\nloss functions and conceptual differences between the two approaches, yet the authors merely emphasised their avoidance\nof generative architectures (such as normalizing flows or GANs) without providing a detailed analysis of the fundamental\nmathematical distinctions in the loss functions. This evasion undermines the article’s innovation: merely transplanting\ndenoising score matching without elucidating its unique theoretical contribution to coarse-grained modelling significantly\ndiminishes the methodological value. While the authors cite an article (Nat. Chem. 2025, 7, 1284-1292) as evidence of\n“extrapolating”, they fail to present comparable data in this paper. If the core advantage of noise-assisted training is reducing\nsample requirements, then a direct comparison should be made between CCG and the present method on identical systems\n(e.g., Chignolin) to determine the minimum dataset size needed to achieve equivalent accuracy. If the authors cannot obtain\nthe CCG code to evaluate the actual system, they should at least compare the efficiency of the two methods on a numerical\ntoy model.\nIn addition, regarding the expansion of application scenarios, the authors’ response to the rare event sampling problem is\nequally unconvincing. Yet the authors merely vaguely mention “can be combined with data generated by enhanced\nsampling” and evade specific case comparisons by citing the lack of all-atom reconstruction methods. In fact, the\nperformance of coarse-grained models in rare events (such as sampling folding transition states) can be evaluated through\nthe statistical properties of their own trajectories, without relying on FCR functionality.\nTherefore, the current revision has not yet met my previous modification requirements, particularly failing to clearly delineate\nthe boundaries between this paper’s methodology and CCG through quantitative comparisons and theoretical analysis. If\nthese analyses cannot be completed, the assertion of “innovation” should be toned down, explicitly positioning this work as\na specific application of denoising score matching within coarse-grained force fields.", "reviewer_2": "I have looked at the response and am happy to see the paper published as is"}, "author_response": "Response to reviewer 1\nIn the revised manuscript, the authors’ modifications and discussions of relevant details are com-\nmendable. The newly incorporated Chignolin system and free energy surface analysis based on\nRg/RMSD enhance the comparability of results. Supplementary Figures S7, S9, and S12 in the SI\nindeed clarify the physical significance.\nHowever,inaddressingthecoreissueImentioned,thatis,thecomparisonwiththeCCGmethod\n(arXiv:2305.01243v1). There are clear shortcomings, suggesting an attempt to sidestep the crucial\npoints.\nReviewer Point P1.1 — Iexplicitlyrequestedacomparisonofthelossfunctionsandconceptual\ndifferences between the two approaches, yet the authors merely emphasised their avoidance of\ngenerative architectures (such as normalizing flows or GANs) without providing a detailed analysis\nof the fundamental mathematical distinctions in the loss functions. This evasion undermines the\narticle’s innovation: merely transplanting denoising score matching without elucidating its unique\ntheoretical contribution to coarse-grained modelling significantly diminishes the methodological\nvalue. Author reply: We apologize for not presenting a more detailed comparison before. We have now made\nthe differences between the two methods more explicit and we have added the following part to the\nmain text (page 12, lines 334-355, highlighted in red):\n“The addition of configurational noise may impact other tasks central to CG MD. For example,\nsubstantial work has recently focused on reintroducing atomistic details into CG protein configurations\nusing ML-based techniques [66-74]. Many approaches produce atomistic configurations that unam-\nbiguously map to their CG counterparts, consistent with traditional bottom-up CG; however, this same\nreconstruction task naturally incorporates additional flexibility between a given atomistic and CG config-\nuration when interpreted with the proposed probabilistic map. Although this formulation has previously\nappeared as a practical relaxation for numerical optimization (e.g., Zhang et al. [69], Pang et al. [70]),\nit can instead be viewed as an exact solution in this probabilistic setting. This compatibility between\nreconstruction and the proposed resolution coupling may in turn improve methods which directly utilize\nsaid reconstruction to accelerate conformational sampling. In this context, recently, Zhang et al. [69]\nproposed learning the fine-coarse resolution coupling from data, and using the learned transformation\nin a generative fashion for efficient sampling in an active learning scheme. However, we note that\nthe full application of these methods to explicit solvent molecular systems often requires complete re-\nconstruction of the all-atom resolution, including the placement of water molecules for each protein\nconformation according to the Boltzmann distribution. To our knowledge, such a reconstruction is still\nabsent from typical procedures. In contrast, the focus of the present work is on the efficient training\nof machine-learned coarse grained models in a data-sparse regime without reference to reconstruction;\nthe learned effective energy function for the coarse-grained system can then be used with physics-based\nmethods (e.g., MD) to sample the conformational landscape and estimate free energies differences.”\nWe also included an additional document for review, in which we make a very detailed comparison\nbetween the two methods in terms of conceptual, mathematical, and application points of view. We\nbelievethatsuchadetailedcomparisonistoospecificforthemaintextortheSupplementaryInformation\nof this manuscript.\nevidence of “extrapolating”, they fail to present comparable data in this paper.\nAuthor reply: While the training of a sequence-extrapolating machine-learned transferable model\nsimilar to the one of the cited manuscript could be a valuable application, we consider it to be out of\nscope of the current manuscript. The goal of this work is not to tackle the long-standing problem of\nchemicaltransferabilityincoarsegrainedmodelsbuttopresentatrainingmethodologythatsignificantly\nimproves the data-efficiency of current machine-learned, coarse-gained models.\nFurthermore, we believe that the Ubiquitin model presented in the revised version of our manuscript\ndemonstrates the extrapolation capabilities of our method. The trained model is capable of sampling\nthe free energy landscape of the protein, including the unfolded state and folding-unfolding transitions,\nwhich are not present in the training data.\nReviewer Point P1.3 — If the core advantage of noise-assisted training is reducing sample\nrequirements, then a direct comparison should be made between CCG and the present method\non identical systems (e.g., Chignolin) to determine the minimum dataset size needed to achieve\nequivalent accuracy. If the authors cannot obtain the CCG code to evaluate the actual system,\nthey should at least compare the efficiency of the two methods on a numerical toy model.\nAuthor reply: Indeed, the code associated with the CCG preprint is not available, nor does the\npreprint disclose any details on the training data of their models and the supplementary information is\nnot available on the arXiv (e.g., there is no mention of the number of datapoints used in the training).\nWe assume they used the full training dataset from the work of Lindorff-Larsen et al. (Science 334,\n517 (2011)) for their Chignolin model. In contrast, we mention in lines 314-316 of Suppplementary\nInformation that in our work we are able to create our MLCG Chignolin model with only 4% of such\ndataset, clearlyshowingtheminimalsamplerequirementofourapproach. Giventheabsenceofausable\nCCG code, a re-implementation from scratch for a more direct comparison is not a reasonable request.\nReviewer Point P1.4 — In addition, regarding the expansion of application scenarios, the au-\nthors’ response to the rare event sampling problem is equally unconvincing. Yet the authors merely\nvaguely mention “can be combined with data generated by enhanced sampling” and evade specific\ncase comparisons by citing the lack of all-atom reconstruction methods. In fact, the performance of\ncoarse-grained models in rare events (such as sampling folding transition states) can be evaluated\nthrough the statistical properties of their own trajectories, without relying on FCR functionality.\nAuthor reply: Wethankthereviewerfortheircommentsabouttheneedtoaddressrare-eventsampling\nsuchasfoldingtransitionstates. WehavenowincludedintheSupplementaryInformationamoredetailed\nanalysis regarding the sampling of transition states. We do this by showing the average fractions of\nnative contacts ⟨Q ⟩ formed in different segments of the protein at each stage of the folding process\ni\n(identified by the global fraction of all native contacts formed, Q). As it can be seen in Supplementary\nFigureS13,ourmodelssamplethefoldingtransitionstatesandthefoldingprocess,formingthecontacts\nin the same order as the atomistic reference (for the proteins for which we have an atomistic reference).\nWe refer to these results in the main text, at page 6 (lines 191-196, highlighted in red):\n“An analysis of the fraction of native contacts shows that the formation order of different structural\nmotifs are well recovered by the models, demonstrating an application in rare-event sampling of folding-\nunfolding transition states (see Supplementary information IIE). We also include in the Supplementary\nas RMSD–R (radius of gyration).”\ng\nAnd also at page 9 (lines 236-237, highlighted in red):\n“At the model is also capable of reproducing the majority of the native contact trends (see Supple-\nmentary Information).”\nReviewer Point P1.5 — Therefore, the current revision has not yet met my previous modi-\nfication requirements, particularly failing to clearly delineate the boundaries between this paper’s\nmethodologyandCCGthroughquantitativecomparisonsandtheoreticalanalysis. Iftheseanalyses\ncannot be completed, the assertion of “innovation” should be toned down, explicitly positioning\nthis work as a specific application of denoising score matching within coarse-grained force fields.\nAuthor reply: We understand the comments from the reviewer regarding the claims about the novelty\nof our work. We consider that with our current reply to the comments and the modifications to the\nlatest version of our manuscript, the differences to the CCG method are shown clearly, and we believe\nthe tone of our claims is appropriate.\n3"}]

Decision letter