Joint Modeling of Spatial and Temporal Multiscales in Molecular Dynamics

The paper lacks evidence that the proposed method solves a relevant molecular simulation task. The task proposed in the ITO paper, which the authors used as a baseline, would be a proper task. This reviewer would consider raising scores if evidence/insights on relevant task can be shown.

Overall, I believe the authors present a promising approach for predicting MD trajectories. However, the following questions should be addressed to clarify key aspects and improve rigor:

1. Soundness of the Method: The authors claim in lines 45-46 that "This spectrum exemplifies how localized spatial interactions correspond to faster temporal dynamics, while extended spatial interactions lead to slower dynamics." Yet, in Section 5.3 Neural Ordinary Differential Equations (Neural ODEs), the same timestep is applied across all ODEs, which seems inconsistent with this principle. Since localized and delocalized components correspond to different dynamical speeds, they should ideally use different timesteps.

   • This oversight also impacts the authors’ claim that the method supports multiscale temporal simulation. In reality, it supports irregular timestep simulation, but the timestep remains uniform across localized and delocalized Fourier components.

   • However, the claim of multiscale spatial simulation is valid.

2. Clarity of Methodological Details: Some parts of the methods section lack clarity and specificity.

   • The training setup for Eqs. 5 and 6 is not provided. Information on the types and dimensions of the neural networks used, as well as training protocols, would be helpful.

   • Eqs. 16 and 17 use the same notation as Eqs. 5 and 6. Do these equations share the same neural networks, or are they distinct? How were they trained?

3. Typo: There is a typo in lines 85-86.

4. Performance Metrics and Representation: The presentation of results and metrics could benefit from adjustments for physical interpretability.

   • I suggest adjusting the x-axis in Figures 2 and 3 to reflect physical simulation time instead of “time scope” or “time frequency,” as these concepts are not well-defined in computational chemistry.

   • Figure 2b shows only marginal improvement over EGNN. Could the authors elaborate on why this is the case? Were there examples where the proposed method underperformed compared to other methods?

   • Figure 2 reports the Mean Squared Error (MSE) of predicted positions, which is a common ML metric. However, it would be more informative to verify whether the method produces physically consistent MD trajectories. For example, the MD17 dataset was generated under NVT or NVE ensembles, which conserve certain physical observables over time. I suggest the authors compute the energies of their predicted atomic positions to assess if the method preserves energy conservation. If energy conservation is not maintained, how does this method compare with others in this respect? I believe this approach would provide a more physically sound measure of prediction accuracy.

• Please correct "ahallengeforementioned"

• Note some recent developments in ML-based surrogate models of molecular dynamics such as Two-for-One (Arts et al 2023), Score dynamics (Hsu et al 2024) and F3low (Li et al 2024)

• What about transferability? Can a model trained on one dataset be used in another? For this paper I am not demanding proof of transferability. But such discussions are necessary.

• The proposed method is deterministic. Can stochasticity be considered/treated?

• Closely related to the above: What's the nature of the MD reactions being modeled? It is downhill relaxations? Then a deterministic model might work. Or thermal fluctuations and barrier crossing? Then this work won't apply. The authors should clearly state the intended application area of their method.

• Are latent feature vectors h and \tilde{h} scalars (rotationally invariant), and x and \tilde{x} vectors? If so, consider stating this explicitly, as \tilde{h} and \tilde{x} are really just latent features of different irreps (0e and 1o) and it is more appropriate to call them so.

• I suggest distinguishing between input features h and latent features \tilde{h}, if that's the correct way to understand it. Please explain what's h.

• I find eq (6) confusing. Why the sum of real coordinates and a vectorial (1o) latent feature. It makes sense to me to do a Fourier transform on vectorial (and scalar) latent features, but including real coordinates x in the GFT looks odd to me, because then you'd have to worry about translational invariance. Please explain how the center of mass (k=0 mode of FT of coordinates) is treated in this work. Physically, because of translational invariance, the k=0 mode shouldn't matter. Otherwise, it makes more sense to me to have the eq(6) without adding real coordinates x.

• Eq (15): It is confusing. Eq (5, 6, 7) says the input features h are fed into the GNN, and \tilde{h} is the output rather than input.

• Does eq (15) suggest the dynamics modeled is time-dependent dynamics? Also in eq (10,11), they are time dependent ODE. Was the ODE explicitly time dependent?

• Eq (7, 16): the term "updated" is confusing. I'd reserve "update" to mean changes from one timestep to the next, i.e. temporal updates. I understand it's meant in a computer science context. But talking about dynamics, it's better to avoid confusion.

• eq (16, 17): are the LHS h and x, rather than the tilde ones? From Eq (18) it appears the tilde symbols were meant to mean both latent features and model predictions. Please rephrase. Strictly speaking eq (16, 17) are just wrong: eq (16) looks like a fixed point evaluation problem but apparently that's not the intention. Eq. (17) similarly means literally the updates are zero on RHS.

• Are the learnable models such as phi and chi in the encoder and decoder the same? If not, please use different symbols.

• How is \Delta T (line 377) defined. Is the the uniform timestep between t1, t2, ...? What's the MD timestep? And please also express \Delta T in real units like ps.

• Table 1: how irregular are the timesteps? How is the related to \Delta T?

• Fig. 2: again please state clearly how much is a timestep. Consider adding a "real-time" x-axis with ps units on the plot.

• Fig. 2: My biggest complaint of this paper is the lack of more meanningful metrics as a MD surrogate model: physics metrics rather than point-wise MSE. Over long time horizons the point wise MSE is irrelevant. What about bond length, bond angle, total energy... distributions? Those are far more important than MSE.

• What is the MSE of a random molecule in MD17 compared with the initial structure? This should provide an upper bound for the results in Table 1.