Thank you for your detailed feedback and your constructive comments. We hope that our detailed responses would solve your concerns.Should you find our reply satisfactory, we kindly encourage you to raise your score.

Q1: More details of the experiments

Thank you for pointing out this. We will add them in our camera-ready version. We now address your concern related to method details you mentioned.

no documentation of the attempts made on the LJ-55 problem

We did not save the results for LJ-55 for some baselines as it is significantly worse than others. A similar pattern has already been observed by iDEM (Fig 9). We note that these baselines will anyway require significantly more energy evaluation times than ours (and PT+DM).

Details about additional hyperparameters in Tab 4

For most of the targets, we tune PT+DM so that the swap rate is ~30%, and we use the same schedule for PT+DM and our approach. We also specifically tune PT+DM on GMM40 as it requires fewer levels. So we in fact put similar effort into tuning the temperatures for both approaches. However, we agree that our approach also has many other hyperparameters to tune, and this can make the tuning more involved. We will update our camera-ready paper to reflect this limitation. We also want to emphysize that more hyperparameters also means a larger design space, which has the potential to improve the performance of our approach further.

More details about the diffusion model used for PT+DM

To ensure fair comparison, we employ exactly the same neural network architecture and training hyperparameters for PT+DM and PTSD.

Q2: The experiments are limited to three tasks

Following your suggestion, we evaluate our model on a more complicated task. As our main focus for the application is molecules, we evaluate our method on Alanine dipeptide (ALDP) in Cartesian coordinates. This task is highly challenging for neural samplers.As we can see in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/ramachandran.png, our algorithm still obtain reasonable performance on this task, which strongly support the effectiveness of our approach.

Q3: The Taylor expansion seems to be a crude approximation when $\Delta T$ not close to 0

We totally agree. However, we found that this crude approximation provides a better exploration performance compared to using the derivative (with auto-diff) directly. We have provide a plot in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/extrapolation_ablation.png to illustrate this observation.

This is possibly due to 2 reasons:

1. The derivative may not be robust, as we only train the model with a finite grid of temperatures;
2. Our approximation eventually forms a auto-guidance, which may have the potential to reduce the network imperfection.

Q4: Figure 1 suggests that PTSD performs better than PT+DM, but Table 1 indicates that it is often outperformed by other methods

We made a mistake in the previous LJ55 results. We now have updated the table in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/updated_main_table.png to reflect this. The energy function evaluation count decreased significantly, while also improving the energy-MMD. Note also the interatomic distance plot, which shows that our method beats BNEM in this regard.

We can see that our approach delivers a performance that is better or on par with other baselines. We also note that even though sometimes the baselines outperform our approach, the improvement is very small. On the other hand, our method consistently delivers good results on these targets, and even works well on the challenging ALDP target.

Q5: no error bars are reported

Thank you for mentioning the missing or error bars, we will update them in our camera-ready version.

Reply to Other Comments Or Suggestions

Thank you for pointing out the inconsistency of the PF-ODE writing, the missing references, and clarity of the VE SDE writing. We will fix them. For the other questions:

How are swaps done?

We use standard PT swap; the acceptance rate is in eq(2) in our paper.

How do we sample MW32 ground-truth?

We follow FAB to first run iid samples from Double Well, and then stack 16 samples together.

Responses to the Questions For Authors

1. We train our diffusion model by standard denoising score matching

2. For CMCD, and other diffusion samplers, it cannot directly train from data without increasing energy evaluation times. In fact, every time we calculate the loss, hundreds of energy evaluations. On the other hand, our pipeline does not need an extra energy evaluation and hence will be significantly more efficient compared to others.

3. We use 10000 samples

Thank you for your review. We hope that our detailed responses would solve your concerns. If any questions or concerns arise, please do not hesitate to let us know, and we will address them properly. However, should you find our reply satisfactory, we kindly encourage you to raise your score.

Q1: the paper is hard to read in general and is applied only to a niche audience

Thank you for your comment. We will be happy to update our paper if you could kindly provide more information in detail about the part that you found hard to read. Additionally,we respectfully disagree with the argument that this paper is niche. Sampling from unnormalized densities is a long-standing task in many areas, including Bayesian inference, Molecular simulation, physics, statistical chemistry, etc. There are many approaches trying to address this problem. Here is a non-exhaustive list of recent advancements: [1, 2, 3, 4, 5].

Q2: The proposed PTSD method is kind of slow in the experiments conducted. Any specific reason for that?

This seems to be a misunderstanding. As we can see from Fig 6 and Tab 2 in our manuscript, our method is actually the most efficient approach in terms of energy function evaluations. It is order-of-magnitude faster than other neural sampler baselines. Could you specify where you found our method slow?

Q3: Proposed method is limited to simple models

Our method does not rely on any specific architecture. One can apply any architecture to our proposed pipeline.

Also, for the problem we are addressing, we believe the model (e.g., EGNN) we use is indeed “SOTA” as it balances well running cost and performance. The architecture we use is a standard choice for the n-body system and small molecules [4, 6, 7]. Therefore, it would be very helpful if you could clarify what kind of model you expect us to evaluate our approach on.

Q4: The experiments in general seem kind of simplistic, and the results in Table 1 are also not that good. Is there a reason for that?

Thank you for your comment. We used the wrong hyperparameter (model size) when we ran the results for Table 1, and hence LJ55 is slightly worse than expected. We have fixed this issue, and we can see the updated results in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/updated_main_table.png, where both the energy-MMD and data-W2 are improved. We also see that number of energy function evaluations of our approach is less than PT+DM across all 3 tasks. The visualization of the interatomic distance in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/LJ55_interatomic.png also reveals that our method does have advantages over BNEM in this regard, and is a strong competitor to PT+DM.

Additionally, we provide results on alanine dipeptide in Cartesian coordinates, which is a challenging task where most of the baselines failed. This significantly highlights the capability of our approach. The experimental results can be found in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/ramachandran.png.

Q5: Paper lacks proof

Our approach relies on Taylor expansion and finite difference approximation for first-order derivatives. There are standard results in calculus. Could you please explain which part you suggest us to include a proof?

[1] Noé, Frank, et al. "Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning." Science 2019.

[2] Doucet, Arnaud, et al. "Score-based diffusion meets annealed importance sampling." NeurIPS 2022.

[3] Midgley, Laurence Illing, et al. "Flow Annealed Importance Sampling Bootstrap." ICLR 2023.

[4] Akhound-Sadegh, Tara, et al. "Iterated denoising energy matching for sampling from boltzmann densities." ICML 2024.

[5] Vargas, Francisco, et al. "Transport meets Variational Inference: Controlled Monte Carlo Diffusions." ICLR 2024.

[6] Hoogeboom, Emiel, et al. "Equivariant diffusion for molecule generation in 3d." ICML 2022.

[7] Klein, Leon, Andreas Krämer, and Frank Noé. "Equivariant flow matching." NeurIPS 2023.

Thank you for your detailed feedback and your constructive comments. We now reply to your concerns one by one. Should you find our reply satisfactory, we kindly encourage you to raise your score.

Q1: Limited empirical evidence & lack of scaling PTSD into more complex problems

To showcase the superiority and better scalability of our method, we conduct experiments on a more complex system-Alanine Dipeptide in Cartesian coordinates, a task that most of the neural samplers failed on. The results of experiments are provided in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/ramachandran.png. As we can see, our algorithm provides high-quality samples on this challenging task, with very few energy function evaluations. (uses almost an order of magnitude less evals than FAB)

We will include this results in our camera-ready version.

Q2: lack of comparison against PIS, DIS, or GFlowNets

Thank you for suggesting these baselines. However, we highlight that PIS, DIS and GFN essentially have the same property as DDS: they mostly use the same model architecture as DDS (parameter the network with a score term); and they require simulating the entire trajectory, which requires hundreds of energy calls to calculate the loss once. Therefore, we use DDS as a representation for this family of approaches.

Q3: Lack of results for FAB on LJ55

The result for FAB on LJ55 is not shown because its training diverged in our experiments. Similar results were also observed in iDEM: From iDEM Fig 9, we can see FBA is significantly worse than iDEM.

Q4: Different results for iDEM on LJ55

We note that in iDEM, they obtain their results on LJ-55 by running a few steps of Langevin dynamics on top of the generated samples. This can be seen from their codes: https://github.com/jarridrb/DEM/blob/main/configs/experiment/lj55_idem.yaml where they set “num_negative_time_steps: 10”. This means they run 10 steps of Langevin dynamics in https://github.com/jarridrb/DEM/blob/main/dem/models/components/sde_integration.py

As this trick is applicable to our approach and all the baselines, we present the results without it.

Q5: lack of logZ, NLL, etc.

Computing logZ and NLL requires model density. Different baselines will have different ways to calculate logZ and NLL: For CMCD and DDS, we need to discretize the SDE into sequence of Gaussians; for diffusion models (ours and PT+DM), we can either use this approach, or estimate it with change of variable on PF-ODE. As different approaches have different discretizations or different underlying processes, this would make a fair comparison more difficult.

In the end of the day, one cares ultimately about the quality of the samples. Therefore, we believe that sample-based metrics are more straightforward and comparable across methods.

Q5: Lack of interatomic distances for LJ

we have provided it in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/LJ55_interatomic.png. We see that PT+DM is a strong baseline for LJ55, but PTSD is a close second in terms of visually matching with the ground-truth distribution. We will add this visualization in our camera-ready version for a more comprehensive evaluation.

Q6: Ablation studies

Following your suggestion, we studied the effect of the amount of temperatures in the GMM task, and report the results in https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/num_temp_ablation.png.

We can see the trend is monotonic: all of the metrics improve with more temperature levels. This indicates we can always use more compute to obtain better results in our method.

Q7: Lack of comparison of the memory cost of training samplers

Our approach encounters a larger memory cost in two places. We now analyze these two costs one by one:

1. When we run MCMC on the samples from the buffer, if we parallelize them, we will encounter a larger memory cost compared to PT. In fact, we view this as an advantage rather than a limitation as this indicates that our method enables efficient parallelization of MCMC into multiple independent chains, with the number of parallel chains determined by the buffer size. On the contrary, standard PT can only parallelize over as many chains as the number of temperatures. The buffer size is normally much larger than the number of temperatures; our approach can, in fact, allow for more massive parallelization.

2. When we sample using our proposed temperature-guided score, we will need to save and call two models together in memory; we may also need to backprop through these two models. In total, they will increase the memory cost to ~4x of standard diffusion model sampling. We agree that this a potential limitation for our approach, and we will include a discussion on this in our camera-ready version.

Thank you for this insightful question. This allows us to realize an additional advantage and limitation of our algorithm, which we believe have made our paper more solid and stronger.

Thank you for your detailed feedback and your constructive comments. We hope that our detailed responses would solve your concerns.

Questions:

Q1: Metrics related to physics

We follow the metrics used in previous literatures, e.g., iDEM, NEM, DiKL, FAB, etc. In these baselines, they do not evaluate on any physical metrics.

One potential metric to use is to estimate the expectation of some manually crafted function using samples generated by our approach, and compare it with the value estimated using ground truth samples. We can view this as a “toy” physical property. In practice, we use a quadratic function of the configuration, i.e., estimate $\mathbb E_{p(x)}[ x^T x ]$. We compare the estimated integral with the ground-truth using the Mean Absolute Error (MAE) and normalise the error as a percentage of the ground-truth integral value. We estimate 90% confidence intervals with bootstrap resampling. As we can see, our method can still deliver good results consistently. We will include this result in the camera-ready version.

Q2: How does the method depend on the number of temperatures?

More temperatures can (1) reduce the burden of training models for each temperature, and (2) can make our temperature guidance approximation more accurate. (1) is because, in practice, we always fine-tune our new diffusion model from the model trained on the last temperature. Using more temperature levels can reduce the difference between levels. For (2), we can look at Eq (11) in our manuscript to see this. When T1 -> T2, the guided score will become accurate. However, similar to PT, running our approach with too many temperatures can make the pipeline less efficient. In practice, we found matching the temperature schedule with that of standard PT can usually lead to a good performance. We provide an additional ablation study for it, by differing number of temperatures for training PTSD on the GMM problem: https://anonymous.4open.science/r/PTSD-rebuttal-icml2025/num_temp_ablation.png.

Q3: Why is the W2 distance better, but the other metrics are not as different between different methods?

This is an interesting observation. We suspect that this is because W2 directly measures the difference between sets of samples, while TVD and MMD are computed on the energy values for the samples, which loses more information. We can also see our approach achieves better results in the above, reflecting a consistent trend with W2.

Q4: Convergence of the proposed method:

one weakness of the method is the lack of "proof" that the algorithm should converge toward the correct probability distribution in some limit.

In fact, we can always get better results by either increasing the number of temperatures or increasing the local parallel tempering refinement steps. We can view these limits as just running annealed Langevin with a model trained along the sampling procedure. In this case, the only error will be the intrinsic model imperfection. Therefore, the convergence is in fact guaranteed.

Other concerns:

Q5: Potential failure case for target distribution with well-separated modes

Thank you for raising this insightful concern. We agree that our algorithm could potentially require many temperature levels for good performance if standard PT suffers due to some phase transition / separated modes. However, we emphasize that this limitation is not unique to our approach - standard PT can also require more chains and require task-specific design (for example, the trajectory PT in the case you mentioned). We will include this discussion in our camera-ready version, and we believe this discussion could make our paper stronger.

Q6: Computational cost

The main limitation might be the burden to train a large amount of diffusion models, when the number of temperatures levels become large [...]

We agree that this can be slow. However, in practice, we always fine-tune our new diffusion model from the model trained on the last temperature. Using more temperature levels can reduce the difference between levels, and hence, the training cost per level will also be reduced.