A Neural Material Point Method for Particle-based Simulations

Thank you for your review, we appreciate the actionable feedback provided, and hope to have addressed the issues you raised. We will include the changes proposed below in a revision within the rebuttal period.

Regarding the weaknesses:

1. Li et al.'s method is a hybrid classic-ML approach. As such it benefits from the strong inductive biases of the physics of the method, but its range of applicability is smaller. On the other hand, NeuralMPM is a fully data-driven emulator, and, as such, it can be applied to a wider range of problems at the price of less powerful inductive biases. For example, NeuralMPM is capable of accurately simulating SPH data (Table 1), a task which we believe would challenge MPMNet as it explicitly implements its physical model.

2. We chose GNS and DMCF as they are well-established, particle-based fully data-driven emulators that assume no knowledge of the dynamics. Another GNN-based method, SEGNN [1], is provided by the LagrangeBench benchmark suite that we used for Dam Break 2D. We also outperform this method and will add it in the appendix.

3. You make a good point. We would like to note that the performance on WaterRamps in terms of MSE is close, and is better in terms of EMD for NeuralMPM than for the baselines. We will update Fig. 4 to show a different dataset with a more convincing result.

4. We are sorry for the confusion. We will create the ground truth and update the figure to show the ground truth for the right hand side particles. We will also clarify the labels.

5. The processor is invariant to the number of particles, but not the whole pipeline (particle-wise operation like position updates still scale like O(N) in both memory and time). In Fig. 6 we show that the FPS remain stable longer than the baselines and the classical simulator, and that the memory consumption is orders of magnitude lower than the baselines. We will update the claim in L469 to clarify the confusion.

6. The result in Fig. 3 shows that NeuralMPM achieves a lower MSE than GNS using the implementation and hyperparameters provided by Sanchez-Gonzalez et al. [2]. We will add a discussion of the discrepancy which we attribute to the hardware they run their model on. Further, we will remove the discrepancy from Fig. 3, as we were not able to reproduce it.

7. This is a good point and we thank you for these suggestions. We will update the abstract and the conclusion to be more precise and better highlight these facts.

8. We thank you for the constructive feedback, and your clarity and willingness to increase the score. We hope to have addressed your concerns below:

9. We will add the comparison in the appendix, using the error over time metric (similar to Fig. 9), and videos for comparison.

10. We have not reimplemented GNS, but rather used the implementation by the original authors, Sanchez-Gonzalez et al. [2]. The implementation in the GNS paper is claimed to have used TPUs, which might explain the discrepancy. We ran NeuralMPM and GNS on the same hardware to ensure a fair comparison.. We will add the TPU clarification in the paper.

11. We will add a section in the appendix, and videos, where we increase the number of particles and present the results, while clarifying the claimed invariance to be limited to the processor and not the full pipeline. Note that this still results in large memory gains.

12. As stated above, we will address this point.

Answers to questions: see comment below

Answers to questions:

1. There are differentiable implementations of classic MPM methods, such as with the Diff-Taichi framework. These are not neural emulators, but rather classical simulators.

2. The input density field and output velocity field present sharp discontinuities (either there are particles, or there are not), which makes it harder to approximate the function with a Fourier-based neural operator.

3. See weakness 2)

4. It might be due to the difference in hardware, even though we trained longer than the original authors. It might also be due to implementation differences, as their public implementation is not the one they used on TPUs (see the bottom line of their README page on Github).

5. We originally did not think it fair, as unlike GNS DMCF did not use Goop in their paper. We have retrained DMCF on Goop (Table 1) and will include a line in Fig. 3

6. Taichi-MPM is the ground truth to which we compare against, it would simply have 0 MSE as we would be comparing a trajectory against itself.

7. They mention it only in passing: "[...] on GPU/TPU hardware, was typically within a few hours for smaller, simpler datasets, and up to a week for the larger, more complex datasets."

8. You raise a valid point. We chose WaterRamps because it was the dataset we used to iterate during the research (more complex than Goop but simpler than the others). It is a legacy decision that we will correct by showing the results on another dataset, as stated above.

9. See answer to sSsx Q2. In addition, Taichi-MPM is the ground-truth classical simulator we use. While it was insightful to include it in Figure 6 for performance, it does not converge to anything as it explicitly defines the physical model.

Overall proposed changes:

• Add SEGNN comparison
• Replace WaterRamps by a more meaningful dataset for Fig. 4
• GT for rectangular domains
• Clarify labels for rectangular domains
• Clarify "NeuralMPM is invariant..." -> Processor is invariant and the pipeline benefits from this
• Update abstract & conclusion with more quantitative details: how many datasets, etc.
• More comparative plots for datasets like Multimaterial, Dam2D and VG to show the gap over GNS/DMCF.
• Clarify the differences/discrepancy with our retraining of GNS vs. the reported results by Sanchez-Gonzalez et al.
• Remove discrepancy in Fig. 3.
• Clarify why we choose DMCF & GNS in the text
• Add line for DMCF in the ablation study

We thank you again and hope to have addressed your concerns and that you will reconsider your score given our answers and the proposed changes.

References:

[1] Brandstetter, J., Hesselink, R.D., Pol, E.V., Bekkers, E.J., & Welling, M. (2021). Geometric and Physical Quantities improve E(3) Equivariant Message Passing. ArXiv, abs/2110.02905.

[2] Sanchez-Gonzalez, A., Godwin, J., Pfaff, T., Ying, R., Leskovec, J., & Battaglia, P.W. (2020). Learning to Simulate Complex Physics with Graph Networks. ArXiv, abs/2002.09405.

Sorry for the delay. I have read your rebuttal thoroughly, but some points have not been addressed thoroughly, in my opinion. I have arranged them in order of relevance.

1. On Fig. 7 and generalisability

I should have stressed this enough, but this is quite crucial to me. My concern is that the evolution of the fluid with time does not seem believable from a physical perspective. Of the two blobs (square and cross-shaped), is hard to believe that the leftmost one is completely missing at time $t_1$. Same idea goes for the right-most figure. Why does the left blob flatten on the ceiling?

The authors have not commented on this. This is also shown in Fig. 19, where Snapshot 1 simply does not look like a believable evolution of the initial condition for basically all the figure shown. The only reasonable explanation to this is that the snapshot has been taken too far in time with respect to $t_0$.

Your feedback on this has been that you will update the figure to show the ground truth for the right hand side particles. However, in Fig. 19 you state that no ground truth is available because the authors of GNS did not provide the physical parameters for the simulation. Now, if you could have simply added ground truth initially to defend the claim of generalisability, why didn't you do so?

I remain skeptical on this point.

2. On Fig. 6

The authors clarify that the processor is invariant to the number of particles, but not the whole pipeline. However, in line 469 the subject of the sentence is NeuralMPM, which I believe refers to the pipeline. In Fig. 6, authors point out that the FPS remain stable longer (i.e., it is more robust as the number of particles increases). This, however, is fundamentally different to invariance, and I believe the claim is overstated.

3. On Fig. 4

I appreciate the use of a different dataset. If DamBreak 2D or Variable Gravity show such significant improvement, it is important to show that.

On the comparison with GINO

I asked for a comparison with Li et al. 2024, but maybe I should have clarified that I was referring to the GINO paper. The authors say: Li et al.'s method is a hybrid classic-ML approach. As such it benefits from the strong inductive biases of the physics of the method, but its range of applicability is smaller.

This doesn't apply to GINO, hence I believe there has been some confusion on this? What paper were you referring to? The authors continue by stating:

On the other hand, NeuralMPM is a fully data-driven emulator, and, as such, it can be applied to a wider range of problems at the price of less powerful inductive biases.

This claim actually applies to virtually any ML based emulator? I don't see this as a novelty of NeuralMPM.

On the intuition on the superiority over FNOs

The author state that FNOs perform worse due to the sharp discontinuities in the input density and output velocity fields. Is this your intuition or can you provide references on this? Also, NeuralMPM discretises particles into voxels, which I assume, from Fig. 1, are fully filled? (i.e. no voxels without particles?). Regardless of a grid fully filled/sparse with voxels, surely your U-Net suffers from the same issue as a FNO?

I remain skeptical on this.

Summary:

I feel the authors have only partially addressed my concerns, leaving me with some doubts explained in the points raised above. Specifically, the lack of clarity on the generalisability claim and GT of the rectangular domains, the lack of proof of the advantages over the proposed U-Net pipeline over FNOs, the lack of comprehensive benchmarking (the authors have brought the compared models from 2 to 3) and the lack of improvement over the reported GNS pipeline (Sanchez 2020) leave me skeptical.

Since I still have doubts on the novelty, impact and on some overstated claims after the rebuttal, I feel that 5 is an appropriate score for the manuscript. I prefer to wait for the final version of the manuscript to finalise the score, to see if the edits in the text make a more compelling argument, especially for Fig 4, 7 and 19.

Thank you for answering, engaging, and clarifying your opinion.

1. In the supplementary material of the paper we have included videos of the rollouts in Fig. 7, showing 10 WaterDrop-XL (larger number of particles) rollouts, and the rectangular rollouts presented in Fig. 7 and Fig. 19. As we said, we cannot add the ground truth used for this figure as we do not have access to the same simulator, physical parameters, or even the generalization datasets from the GNS paper (we contacted the authors and asked for them, but they said that open sourcing the generalizaion datasets was outside of the scope of the project). What we proposed was to generate a new dataset with a simulator to which we have access, and do the experiments on that. We will not be able to do that within the rebuttal period, as our access to compute is limited. However, we once again stress that the videos of Figs. 7 and 19 are already in the supplementary material, showing the full generalization rollouts.

2. We will revise the claim of invariance to clarify it is for the processor. However, we note that our method is orders of magnitude more memory efficient than the baselines, and ask that this be taken into account. Further, both p2g and g2p scale linearly with the number of particles.

3. We have updated Fig 4. and added similar figures for other datasets (including the old Fig. 4) in the appendix. Further, we have included videos of rollouts on all datasets of all baselines, NeuralMPM, and the ground truth that visually shows the difference. Further, in Figs. 2 and 3 of the rebuttal PDF (9 and 10 in the appendix of the paper) we show the rollout error (MSE and EMD) as a function of time for all datasets (including DamBreak2D and VariableGravity) and baselines. The advantage of NeuralMPM is clear in these plots.

GINO: Thank you for the clarification and sorry for the confusion. GINO introduces a NO on irregular grids thanks to a GNN-based learned transformation from irregular to regular grids. Their work within the context of Eulerian simulations is impressive. The main difference between our two methods is the context, we work on a dynamic point cloud, requiring graph reconstruction at every time step. This requirement is one of the limitations of graph-based simulators and one of the reasons for GNS slowness and memory inefficiency compared to NeuralMPM. On the other hand, NeuralMPM has linear particle aggregation complexity. GINO does this via a linear O(N) method, similar to NeuralMPM. However, our irregular to regular mapping (p2g) does not have any learnable parameters and requires no training. Moreover, we have applied NeuralMPM to Lagrangian point clouds, and have shown its advantages to other emulators that target the same domain. The DMCF baseline uses Open3D for the graph construction (same as GINO), and we have shown to be faster and more memory efficient than it.

The claim about NeuralMPM and inductive biases, as you say, is true for almost all ML-based emulators. This is what we meant by "NeuralMPM is a fully data-driven emulator". We do not claim to have introduced this advantage, rather it drives from being data-driven. Hybrid simulators, which we discuss in L132-137, do not fully enjoy this advantage.

FNO: We want to highlight that the particular choice of processor is not our main contribution, rather the whole pipeline is. In our experiments, we found the UNet to be better than the FNO but that does not mean NeuralMPM must always be used with a UNet processor, nor do we claim the UNet to be a better architecture than the FNO in general.

That said, we will give our intuition for why the UNet performed better in our experiments. The voxels that have no particles are empty, and therefore have 0 density and 0 velocity. The FNO operates only on lower frequencies, removing the higher ones (Fourier Neural Operator for Parametric Partial Differential Equations, Li et al. Section 4 and Fig. 2). This can make modeling sharp discontinuities hard. That is simply our intuition, and we do not have a reference for it. We once more highlight that the comparison of a UNet and an FNO processor is not the main contribution of our work, nor is it generally applicable to all problems.

We have provided videos of the full rollouts of Figure 23 in the supplementary material ("Rectangular" folder, both in the zip file and in the drive mentioned in the general comment, link here https://drive.google.com/drive/folders/1lj8a-iHoAXOyT3HFl5mXa0Bc5_nHbdZb). They look realistic and believable, which is why we claim generalizability.

In the same zip and drive, under WaterDrop-XL we provide rollouts for another generalization task, one which was also used in the original GNS paper. We train the model on WaterRamps and do the test rollouts on WaterDrop-XL, a set with more particles, more energy (i.e. higher velocities), higher density, and longer simulation. We beat a GNS model trained and used in the same way (Figure 11) in terms of the MSE. We will also provide full rollout videos for NeuralMPM and GNS on this dataset, where the advantage of NeuralMPM is clearly seen.

In the supplementary material (zip and drive) we have provided full rollouts for all sims on all datasets with the ground truth, NeuralMPM, DMCF and, GNS shown together. The figures in the paper are snapshots from these videos, but the videos provide a much more detailed context and they need to be taken into account.

Following the author's response, and having noticed the changes provided to the manuscript, I decide to retain my score of 5. The motivations behind this choice below:

Fig. 4. I appreciate you changing it to a more descriptive dataset and adding further examples in the appendix. However, I feel the original improvement over other datasets is slightly overstated: while VariableGravity shows a clearer improvement with respect to other methods, it seems not that different with respect to the original WaterRamps. Results in Table 1 seem misleading, considering that, while the MSE is much lower, visually there's little difference. I have looked at some of the video comparisons, too, for the WaterDam 2D dataset, and improvements seem marginal.

Fig. 3. Have you retrained the GNS? You say that the dotted orange line (2.4 ×10−3) indicates the MSE we obtained for GNS after 240 hours (20M training steps).. If I am not mistaken, in the previous version of the manuscript you had reported the results they obtained in orange, which were consistently lower than your ablations (and that you have removed in the revised manuscript), and in a different colour you reported your re-implementation. The original version of the manuscript could have hinted at the fact that GNS, if reimplemented, might not perform as well as originally reported, but by further tuning you could reproduce those results, since the orange line now is much closer to what originally reported.

Generalizability: I feel the authors haven't tackled this issue. I have looked at some of the WaterDrop-XL dataset videos, and I feel they confirm my initial skepticism. There's some artifacts that don't seem realistic physically, such as water blobs deforming as they come down or that they "slide" along the side of the domain. Now, this dataset is not that different from those presented, as the number of particles is larger but the domain is similar. What is happening in Fig. 7-21-22 is likely a similar phenomenon - the behaviour is quite unrealistic, especially considering that the domain is much more different than those of training data. I think these observations are quite obvious to an external person, since such figures simply don't show a believable evolution of the fluid with time. Considering the lack of ground truth, the claim of generalisability is, at best, overstated.

I appreciate that conducting further experiments might be hard. I was not expecting that, but rather an honest explanation of that behaviour.

On benchmarking: I feel you're being imprecise. You have reported results of 17 configurations of 3 models, one of which is your own, and one of which has been published 4 years ago and, based on the original Fig.3, seems to still outperform NeuralMPM. Now, implementing 17 different models is a different thing than 3 models with different parameters, and it does not require the same amount of effort. I would have been happy simply with a deeper search of the literature.

I understand that NeuralMPM has other advantages over GNS, and I appreciate the efforts put in the rebuttal, including the authors' clarification on GINO and FNO. However, several claims made, including the generalisability over more particles/rectangular domain, the large improvement over past methods and the invariance with the number of particles, seem quite opaque, even after the revision.

Having considered these facts, the lack of extensive benchmarking and of solid novelty, I would consider this paper below the acceptance threshold. I believe this is fair and conservative, also considering other's reviewers harsher opinions on this work.

I apologise if I am repetitive, but I'll mention this a fourth time:

Current Fig. 23, rows 2, 3, 4, 5. Does that seem a believable evolution of Snapshot 1 given the Initial Condition on the left? Given your answer to this question, and considering that no ground truth is available, can you claim generalisability?

Thank you for your review and comments, we will address them below.

1. The works you have suggested are contemporaneous with ours, so a comparison is naturally impossible. They are hybrid methods, on which we have commented in L132-137. In summary, these approaches achieve impressive results by leveraging extensive physics knowledge, but this reliance also limits their applicability. Fully data-driven methods, like NeuralMPM, are more general and can be applied to a wider range of problems.

2. Yes, you are correct. Replacing the processor (in our case a UNet) with a 3D counterpart will extend the method to 3D. We have some early results for that that we will publish in a follow-up paper.

3. We have provided such analysis in Fig. 6, showing that NeuralMPM is faster and more memory efficient than the baselines and the ground truth MPM simulator.

4. Time integration is controlled by the time bundling $m$ parameter. In Fig. 3 we study the effect of $m$ on the accuracy of the model. It can be seen there that it can operate at a time step 8x-16x that of the ground truth data. In turn, the ground truth simulator operates at a much lower time step, making this advantage even more pronounced.

We hope to have addressed your concerns and that you therefore will consider raising your score. Thank you for your time and effort in reviewing our work.

Thank you for taking the time to review our paper.

Concerning the weaknesses section:

1. s you say, autoregressive model stability is a challenging problem. Addressing it is an active field of research on its own right [1] We chose to decouple the framework used for the neural emulator from the rollout stabilisation mechanism by using a common technique, autoregressive training [1, 2]. In Section 4.3 (L470-L478) we produce longer rollouts of 1000 timesteps. To strengthen this argument we will provide a video of longer simulations where we add water particles.

2. That depends on the architecture of the processor used. Both the U-Net and FNO backbones can process different grid sizes without retraining. As shown in Fig. 7, the U-net backbone can extend to larger domains (and thus different grid sizes), but might fail at a finer grid in the same spatial extent, due to fixed kernel sizes. However, using an FNO backbone would support updating grid velocities over different grid resolutions without retraining, thanks to its operator nature [3]

3. While it is true that 3D requires more memory and training time, we already have preliminary results that are less expensive than other data-driven methods, especially GNS which we could not retrain on 3D data on 32GB-VRAM GPUs. We plan to present those results in a future work, as we believe they do not add to the discussion here, and require time consuming experiments.

Answers to questions:

1. We use the reference implementation provided by Taichi (see https://github.com/taichi-dev/taichi/blob/master/python/taichi/examples/simulation/mpm128.py), based on MLS-MPM [1]. We want to stress the fact that our method is not a direct comparison to MPM, but rather a data-driven approach, where MPM (and SPH), was used as ground truth. Our goal is not thus to compare to MPM implementations.

2. In Fig 6. we limit-test NeuralMPM, the baselines, and the ground truth simulator Taichi-MPM in terms of speed and memory. We did not test the accuracy of the simulations for any of the methods.

3. This line refers to rectangular domains, such as a larger non-square domain. A circular domain can be achieved by inscribing the circle in a square and placing the appropriate boundary conditions.

4. Most of the simulations have non-zero initial velocities, this can be seen in the videos of the supplementary material. We do not have smoke simulations.

5. • We trained a different model for each dataset. However, the model trained on MultiMaterial can handle simulations from WaterRamps, SandRamps, or Goop. The MultiMaterial dataset itself contains single-material simulations.
   • Different properties can be embedded into the model, such as different fluid types (e.g., Multimaterial), but the model will be limited by its training data, except if the physical parameter is embedded into the model directly, like we did in VariableGravity.
   • Boundary conditions are naturally handled through the use of static "wall" particles for borders and obstacles. This allows for very general boundary conditions.
   • Initial velocity is also naturally handled as NeuralMPM is conditioned on a single previous state. Most of the simulations have non zero initial velocities.

6. To be differentiable, the whole implementation (i.e. each operation) must be differentiable, and simulators are complex and expensive to craft, sometimes requiring reformulations of the dynamical models themselves. Further, the computation of the derivative has to be efficient, which is a non-trivial challenge for complex or long simulations.

7. Figure 1 depicts the end-to-end architecture of the network, from the input point cloud to the m output point clouds. The processor part (i.e., the neural network) is not depicted in detail as any image-to-image backbone can fit, and we do not consider this to add meaningful information to the description of the pipeline.

We thank you again for your review, and have proposed the following changes to address your concerns:

• Longer simulations where particles are added.
• New results showing the generalization to different domains, with ground truth.
• Add comparison videos for the baselines

Having taken the perspective provided by our answers, and the proposed changes into account, would you kindly consider raising your score, or provide us with further actionable feedback?

References [1] List, B., Chen, L., Bali, K., & Thuerey, N. (2024). Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamics.

[2] Prantl, L., Ummenhofer, B., Koltun, V., & Thuerey, N. (2022). Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics. ArXiv, abs/2210.06036.

[3] Li, Z., Kovachki, N.B., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A.M., & Anandkumar, A. (2020). Fourier Neural Operator for Parametric Partial Differential Equations. ArXiv, abs/2010.08895.

Thank you for your review. We have addressed your questions and concerns to the best of our ability below. While we value your feedback and appreciate constructive critique, we find it difficult to identify clear actionable steps based on the points raised. Could you clarify the reasoning behind the rejection and whether our explanations below have influenced your perspective?

For the weaknesses

1. • Our purpose is not to develop a MPM competitor but rather to build a simple learnable model, inspired by the structure of MPM in order to inherit some of its advantages. The model does benefit from the MPM pipeline (particles -> grid -> particles) yet remains simple. We have adapted some of the solutions used in MPM to problems in the neural emulation field (e.g. the problem of extracting information from the point cloud L57-L62, L107-L110), being novel to the best of our knowledge.

   • The time bundling parameter $m$ is a hyper-parameter which can be tuned, depending on the training data. $m$ can be set to 1, leading to a 1-step-forward autoregressive model. Ablations of this hyper-parameter are shown in Fig. 3 and Section 4.1

2. • We note that we did not generate most of the data, and instead used the datasets provided by Sanchez-Gonzalez et al., as they represent a broad range of physical interactions. Further, we also have an SPH dataset (Dam Break 2D), where NeuralMPM outperforms the baselines. To further show we are not limited to the MPM domain we will add another non-MPM dataset.

   • The number of particles for each dataset is reported in Table 1 (second column, "N"). Note WaterDrop-XL goes up to 10k. The third column is the temporal duration in the number of frames. The duration in seconds (i.e., depending on dt), can be found in the original dataset references. The Dam Break 2D dataset uses dt=0.03 (i.e., 12s-long), while the all other datasets use dt=0.0025 (i.e., from 1 to 2.5s-long), which is often long enough to reach a stable state, considering the velocities.

3. • We strongly disagree that the evaluations are insufficient. We train the model on 6 different dataset (WaterRamps, SandRamps, Goop, MultiMaterial, VariableGravity, Dam Break 2D), comprised of different materials and dynamics, and compare its performance on those dataset against well-established data-driven baselines. Further, we evaluate generalization quantitatively and quantitatively (WaterDrop-XL) on one dataset and qualitatively on another (Large rectangular domains). We also showcase that the model’s differentiability can be used to solve inverse problems. Moreover, the method can be trivially extended to 3D domains, by replacing the processor with a 3D counterpart. However, 3D training requires more memory and training time (compared to the 2D case), making extensive experiments very time consuming, for both NeuralMPM and the baselines. For example, we could not even run the GNS code with 3D data on a high end GPU (Telsa V100, 32GB). We have preliminary 3D results, and plan to present them in a future work, but believe they add little to the discussion here.

4. We acknowledge that evaluating against ground truth is feasible but was not done mainly to reuse the same model (i.e., trained on WaterRamps) as used for WaterDrop-XL. We agree this is not ideal, and will create new simulations that will be extended the same way, with corresponding ground truth.

5. We thank the reviewer for this good remark, we will add videos for comparing against the baselines visually.

6. In L25 we said we achieve “comparable or superior long term accuracy” compared to other methods, namely GNS and DMCF.. This is shown in Table 1, where NeuralMPM has lower MSE on almost all the taks, and lower EMD on all of them.

7. Thank you for bringing this work to our attention. We will add it in the relevant work section.

As for the question

1. GNS indeed adopts a much smaller batch size, as only a size of 2 fits on a V100 GPU with 32GB of RAM. In addition to that, the method is slower due to the graph construction, which becomes quite costly for increasing numbers of particles. In contrast, NeuralMPM, being more memory efficient, can fit much larger batches in memory, and bypasses the expensive graph construction.

We thank you again for your review, and have proposed the following changes to address your concerns:

• Add another non MPM dataset.
• New results showing the generalization to different domains, with ground truth.
• Add comparison videos for the baselines

In light of our answers and the proposed changes, would you raise the score, or provide additional actionable input and elaborate on how you have arrived at the decision to reject?

Thank you for your review. We answer your questions and issues to the best of our ability below. We appreciate the feedback and value constructive critique, though we find it challenging to identify actionable steps based on these arguments. Could you clarify the arguments leading to a rejection, and whether your perspective has changed given our explanations below?

Now, to answer the concerns you raise in weakness section:

1. Our method aims to emulate and accelerate existing simulation approaches by learning directly from data, rather than achieving the most accurate fluid simulation. While NeuralMPM is inspired by and incorporates elements of the MPM framework, it is data-driven, allowing it to effectively adapt beyond the usual MPM domains. This includes emulating simulations for incompressible fluids and SPH data (as shown in Table 1). That is, with adequate training data, the inductive bias introduced by MPM can be "overwritten". Additionally, we demonstrate that NeuralMPM outperforms fully Lagrangian data-driven methods like GNS in accuracy and stability.

2. We acknowledge throughout the paper that our method is based on the classic MPM and therefore benefits from its advantages (L52-L64). In L519-521 we do not claim that those advantages are unique to NeuralMPM or introduced by us, rather, we are claiming that NeuralMPM has those advantages over the baselines we compared against. We consider inheriting the advantages of MPM a strength, compared to other data-driven methods.

Regarding your questions:

1. The motivation stems from the computational limitations of surrogating gather operations. Any ML surrogate would have at least O(N) complexity, the same as scatter and gather operations. A tree based GNN surrogate, for instance, would require O(N log N) computations to build the graph, without taking into account additional operations, the costly training procedure, or the extra error introduced by the surrogate. [3-4] are examples of this.

2. On WaterRamps and SandRamps, there is a performance gap between NeuralMPM and GNS in terms of the MSE, but NeuralMPM has better EMD. We believe that the MSE gap is due to the GNS hyperparameter tuning, and that it can be reduced with a more extensive hyperparameter search for NeuralMPM. In VariableGravity, we used both the original GNS hyperparameters and ran an extensive hyperparameter search (60 GPU days, L255) to optimise GNS performance and ensure fair comparisons, while for Dam Break 2D we used the model provided by LagrangeBench [2], but retrained it for 20 million steps (like GNS) instead of the provided checkpoint, trained for 500,000 steps. For the MultiMaterial case, the sharp boundaries between materials are challenging to model in free space but are more naturally represented on a grid. Our intuition for the better convergence of NeuralMPM is that the combination of the voxelization and an appropriate neural architecture (UNet, FNO...) allows the model to easily capture global structures and interactions, instead of having to rely on numerous message passing steps (L107-L110). It also frees model capacity as the spatial structure is presented in a more compact way and the model does not have to infer it from the point cloud directly (L57-L62).

3. NeuralMPM can simulate a larger number of particles without retraining. We present such results in Section 4.3 (L470 - L478). In particular, we evaluate a model trained on WaterRamps (2.3k particles approximately and 600 timesteps) on WaterDrop-XL (7.1k particles approximately and 1000 timesteps). Some of the WaterDrop-XL simulations have up to 10k particles, an almost 5x increase in the number of particles. In Fig. 6, we limit-test the speed and memory of both NeuralMPM and the baselines, without retraining.

We will add a section in the appendix with generalization to progressively larger grids, until it fails.

We thank you again for your review. We propose to improve our generalization experiments with the larger grid until it fails. With the perspective provided by our answers, could you provide additional actionable input, or elaborate on how you have arrived at the decision to reject?

References

[1] Wang et al, A Massively Parallel and Scalable Multi-GPU Material Point Method, ACM Trans. Graph (2020)

[2] Toshev et al. (2024). Lagrangebench: A lagrangian fluid mechanics benchmarking suite. Advances in Neural Information Processing Systems, 36. https://github.com/tumaer/lagrangebench

[3] Hao, Z., Ying, C., Wang, Z., Su, H., Dong, Y., Liu, S., Cheng, Z., Zhu, J., & Song, J. (2023). GNOT: A General Neural Operator Transformer for Operator Learning. ArXiv, abs/2302.14376.

[4] Li, Z., Kovachki, N.B., Choy, C., Li, B., Kossaifi, J., Otta, S.P., Nabian, M.A., Stadler, M., Hundt, C., Azizzadenesheli, K., & Anandkumar, A. (2023). Geometry-Informed Neural Operator for Large-Scale 3D PDEs. ArXiv, abs/2309.00583.

Thank you for your response, and for clarifying the reasons behind your decision. We appreciate the concrete feedback you provided and we will clarify some points and address first the weaknesses, then the rest.

We are sorry for the confusion. In the statement "with adequate training data, the inductive bias introduced by MPM can be 'overwritten'", we wanted to stress that our method is not limited to the domains of traditional MPM, but can be applied to other domains as well, although we expect it to be especially well suited to traditional MPM domains. Regarding "which indicates the proposed method can actually outperform MPM with regard to accuracy given good SPH data," we clarify that MPM and SPH are fundamentally different dynamical models, each with unique strengths and weaknesses. Our point was that, while MPM cannot replicate SPH-like simulations due to its construction, our method can, when trained on SPH data.

In Fig. 6, we show that our method is faster than the MPM ground truth (acceleration). With regards to the enhancement, we do not claim to be superior to traditional methods, we claim to be faster, and to be able to solve inverse problems (Section 4.4) which we consider an enhancement. Importantly, we claim and show to be faster, more accurate, and have better memory scaling than other data-driven methods.

We will reformulate our goal explicitly: "Our method aims to emulate and accelerate existing simulation approaches, while being superior in terms of speed, memory, and accuracy to existing data-driven methods, and being able to solve inverse problems.". That is, our main contribution is the improvement upon existing data-driven baselines.

(Points 1 and 2) We support our claims for generalization from 2.3k particles to ~8k in Section 4.3 and Fig. 7. We do not claim that our method generalized from 5k to 30M particles, nor do any of the baselines. In Fig. 6 we tested the throughput and memory of the untrained models (baselines and NeuralMPM). We did not see any difference between the converged models and the unconverged ones, for neither the baselines nor NeuralMPM, as the particles are always kept within the domain by construction (no large numbers or exploding quantities). We will make this explicit in the revised version.

(Point 3) We did not run the same simulator as Sanchez-Gonzalez et al. (2020), as it has been unmaintained for years and we were not able to compile it (https://github.com/yuanming-hu/taichi_mpm). We did have access to a subsequent version, but we were not able to replicate the dataset. We note that there is a video in the supplementary material showing various realistic simulations. However, as further evidence and as told by the other reviewers, we will create ground-truth data and show it explicitly using an SPH simulator.

We hope to have addressed your concerns, and will add ground truth for generalization within the rebuttal period. We kindly ask you to reconsider your decision in light of our responses, especially the improvements over data-driven methods.

Thanks for the reply. I appreciate the authors clarify the motivation and claims. However, some points for supporting the claims are not solid enough: 1) accelerate existing simulation approaches, in Figure 6, the authors compare their untrained model to the traditional MPM, it is not convincing since it is unclear whether the trained model would diverge the simulation or not. Diverged/undiverged simulations have very different particles distributions, which affects the performance dramatically. Even the simulation is not diverged, the performance between a highly dynamic and quasi-stationary simulation is also very different. The authors would better compare their trained and well-behaved (i.e., can output reasonable simulation results) model to traditional simulators. Otherwise it is not a fair comparison as it is actually just running the low level CUDA operators but not the proposed method in a simulation context. 2) being able to solve inverse problemsThe ability to solve the inverse problem is not an enhancement as there are already efficient differentiable MPM solvers several years ago (difftaichi [1]) there are also bunch of downstream applications using the diff mpm such as plasticinlab[2], PAC-nerf[3], PhysGaussian[4] and many more. The cases shown in these previous work are much complex than the one shown in the paper. Therefore it is not convincing to state this point as a claim without any comparisons to existing differentiable MPM solvers. I suggest that the authors revise the current manuscript and submit to another venue.

[1] Hu et al, Differentiable Programming for Physical Simulation (ICLR 2021)

[2] Huang et al, Plastcinelab: A soft-body Manipulation Benchmark with Differentiable Physics (ICLR 2022)

[3] Li et al, PAC-NeRF: Physics Augmented Continuum Neural Radiance Fields for Geometry-Agnostic System Identification (ICLR 2023)

[4] Xie et al, Physics-Integrated 3D Gaussians for Generative Dynamics (CVPR 2023)

Dear reviewers and area chair,

We thank all the reviewers for the discussion so far. We have uploaded the revised version of the paper, with changes highlighted in red. In the (updated) supplementary materials, you will find new rollout videos showing the baselines, NeuralMPM, and the ground truth for all experiments, as well as the generalization rollouts for NeuralMPM. Due to size constraints, some are in the supplementary materials and the rest are in this anonymous drive https://drive.google.com/drive/folders/1lj8a-iHoAXOyT3HFl5mXa0Bc5_nHbdZb. This will help the comparison and support the accuracy claims of NeuralMPM in Table 1. A non-exhaustive summary of the changes we made in the revised version is as follows. Further details are given in the response to the reviews below.

1. We propose changing the name of the paper to "A Neural Material Point Method for Particle-based Emulation" to better reflect our contribution.
2. Reworded the abstract and conclusion to be more quantitative.
3. Updated Fig. 4 and added similar figures for the other datasets in the appendix.
4. Updated Fig. 6 caption.
5. Added Fig. 9 and and Fig. 10 in the appendix.
6. Added videos of all rollouts in the supplementary materials.
7. Reworded the invariance claim to be about the processor, and added the linear scaling of p2g and g2p.
8. Added an extra comparison in the appendix against a different implementation of GNS and SEGNN (Brandstetter et al. 2021), as provided by LagrangeBench (Toshev et al. 2024). Our conclusions remain the same.
9. Added Figure 11 in the appendix to show that NeuralMPM performs better than GNS at the WaterDrop-XL generalization task, despite using a model with a slightly lower MSE.

In addition, given how the discussion phase is unfolding, we feel compelled to address, once again, a fundamental misalignment in how our work is being evaluated. NeuralMPM should be assessed as a data-driven emulator that learns from simulation data, not as a physics engine competitor. This distinction is crucial. We strongly emphasize that our main contribution and claim is not superiority to traditional numerical methods, nor an improved version of MPM. We do not claim that in the paper. Rather, we claim and show to strongly outperform the neural emulator baselines. Previous works in this space (GNS, DMCF) were evaluated based on their ability to learn from and reproduce simulation data, without requirements to validate against multiple simulators or justify every observed physical pattern. They showed results on similar datasets with similar evaluation protocols. Our work substantially improves upon these methods in terms of:

• Training speed: 15h compared to 10 days (i.e. 6% of the training time)
• Inference speed: 5x-10x faster
• Memory usage: 10x-100x less
• Accuracy: superior EMD for all datasets, superior MSE for all datasets except 2, where it is very close.
• Multi-material interaction (MultiMaterial dataset, see videos in the supplementary material)
• Inverse problems (while both GNS and DMCF are in principle differentiable, they do not show results on inverse problems)

The additional requirements being suggested - such as validating against multiple simulator types, going 3D, justifying every observed flow pattern, asking for physical realism beyond the capabilities of the underlying simulator (look at the ground truth of some of the videos in the supplementary material where particles stick to the walls, which was simulated using the well established Taichi-MPM) or demanding reimplementations of previous data-driven emulators over official codebases - go well beyond what was expected of previous work in this space and create unreasonable barriers for research progress (see e.g https://openreview.net/forum?id=6niwHlzh10U for the reviews of DMCF). Evaluating NeuralMPM as a physics engine replacement is not only unfair but also misses the key contributions of our work.

As discussed all along, we are happy to improve clarity around our claims and positioning further -- and we thank the reviewers for their feedback in this regard -- but ask that the work be evaluated within its intended context as a machine learning contribution advancing the state-of-the-art in neural simulation emulation.

The authors.

The drive folder now includes updated videos of the WaterDrop-XL rollouts, showing the ground truth, NeuralMPM, and GNS for comparison. The GNS rollouts are worse, supporting the results from Fig. 11. Note that both modles, GNS and NeuralMPM, are the ones reported in Table 1 on WaterRamps. GNS has slightly better MSE than NeuralMPM on that dataset. Despite that, NeuralMPM generalized better.