Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper suggests a novel inductive physical bias to guarantee the conservation of momentum. The latter is achieved by reconsidering the resampling strategies and a reformulation of a continuous convolution operation to incorporate the physical knowledge as a hard constraint to respect the symmetry of interaction of particles by construction. 
strengths_and_weaknesses: This work is inspired by Lagrangian mechanics where fluids could be seen as a constellation of particles in interaction. Particles are represented as nodes and their interactions are represented by edges. This construction is a good starting point to make the analogy with Geometric Deep Learning (GDL) models, including Graph Neural Networks (GNNs) and points of cloud. In GDL, we distinguish isotropic and anisotropic convolutions which are widely used and well studied. However, when it comes to design a continuous convolution operation for physical problems one should look carefully at the sampling strategy (dense / sparse connectivity at different local regions), the symmetries present in the data at different physical scales and how to design a smooth kernel that propagates correctly the physical signal while preserving its physical properties. 
In this study, the authors proposes a novel design of a convolution operation capable at encoding the conservation law as a hard constraint relying on a well studied continuous operation, know as "CConv" instead of incorporating this information in the loss function as soft constraint. One of the strength of this hard constraint is to physically guide the neural network in the learning process for better generalization which a crucial topic in physics. 
The proposed convolutional kernel is smooth by construction and is derived from concepts in Lagrangian discretization, mainly Smoothed Particles Hydrodynamics to integrate in hard the symmetries of the underlying physical problems. This way of designing a kernel allows to conserve the physical properties "symmetries" by design while keeping the properties of convolutional operation by design, namely permutation invariance and translation equivariance. In addition to, a careful choice of sampling scheme is taken into consideration which plays also a crucial role as the choice of convolutional operations.

The weaknesses in this paper are related to the lack of comparison with challenging datasets as done in the related works, including for instance large scale meshes with challenging obstacles and geometries to show the versatility of the proposed model. Moreover,  it could be valuable and insightful to study the robustness of the proposed convolutional operation to different discretization schemes, sampling strategies and noisy data since the goal is generalization.

In general, the paper is well written, clear and easy to follow. The claims and their structure are quite clear and the mathematical variables are well defined. 
The study of the related works is well done. Notably, the authors makes a relationship between different concepts in physics, machine learning, signal processing and computer vision. 
The experimental study is fair and convincing even if it could be extended.

questions: 1/ What is alpha in equation (3) ?

2/ Why the restriction of P_D=P_Q ? and what is its implication on the expressive power of the model ?
limitations: This work doesn't have societal concerns and the authors are aware of the current limitations of the proposed work.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The authors present a physics-guided deep learning model for fluid simulations. The method uses continuous convolutions to process particle representations of fluids where the convolutional kernels in the final layer are constrained to be antisymmetric. The antisymmetric constraint enforces conservation of momentum. The authors show that embedding this constraint results in improved simulation results compared to other methods.
strengths_and_weaknesses: - Originality: The proposed method is an application of continuous convolutions to particle representations of fluids. The proposed intervention is to constrain filters in the final layer to be antisymmetric. The related work is adequately situated in the context of existing literature, I encourage the authors to consider adding the literature on Lagrangian neural network to related work(see questions/comments).
- Quality: The work is technically sound and the claims are well supported. A limitation implicitly noted in the conclusion is that the learned representations are not rotationally invariant.
- Clarity: For me the paper was not easy to read. E.g., I think the paragraph at [163] interrupts the flow, perhaps it is better placed at the end of the subsection. It may not be needed since my sense is the same arguments are given in the related work section. There were also notational clashes and abreviations that were never written out (see questions/comments)
- Significance: The paper addresses the issue of learning physically plausible fluid simulations which is a challenging problem with the potential to facilitate scientific discovery.

questions: - Possible typos:
  - Line 31: write out SPH
- Questions:
  - WBC-SPH: Are the strength and direction of gravity determined randomly?
  - [261] : [EMD] is especially important to reliably evaluate long simulation sequences. What is the intuition here?
  - Figure 1: In eq 2 G is a function of x_t and v_t in the figure it is a function of x’_t, v’_t.
- Comments:
  - Related work doesn't include other NN + Lagrangian methods [1][2][3][4]
  - [116] G is used to denote both the linear momentum and the network that predicts the update which is a bit confusing
  - [31] SPH is not defined until the next section
  - [219] F is used for features and forces which is a bit confusing
  - [318] “An adapted variant of PointNet” – how is it adapted?
  - [320] “For fairness…” It seems more appropriate to use the dataset of CConv since they are using a more similar methodology (CConv vs GNNs)

[1] Saemundsson, Steindor, et al. "Variational integrator networks for physically structured embeddings." International Conference on Artificial Intelligence and Statistics. PMLR, 2020.
[2] Lutter, Michael, Christian Ritter, and Jan Peters. "Deep lagrangian networks: Using physics as model prior for deep learning." arXiv preprint arXiv:1907.04490 (2019).
[3] Cranmer, Miles, et al. "Lagrangian neural networks." arXiv preprint arXiv:2003.04630 (2020).
[4] Sanchez-Gonzalez, Alvaro, et al. "Hamiltonian graph networks with ode integrators." arXiv preprint arXiv:1909.12790 (2019).

limitations: A limitation implicitly noted in the conclusion is that the learned representations are not rotationally invariant. I suspect rotationally symmetric kernels will substantially reduce expressivity. It may be possible to use something like group convolutions [1].

[1] Cohen, Taco, and Max Welling. "Group equivariant convolutional networks." International conference on machine learning. PMLR, 2016.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper proposed the Antisymmetric Continuous Convolution that can guarantee linear momentum in learned physics simulation, and the authors demonstrated the effectiveness of this design on several particle-based simulation datasets. 
strengths_and_weaknesses: $$\textbf{Strengthes:}$$
Overall, I enjoyed reading this paper. The writing is very clear and the proposed idea is quite neat. The ablation study of the model architecture really strengthens the paper. 

$$\textbf{Weaknesses:}$$
1. Lack of Baselines: I think the more SOTA models for forecasting fluid dynamics on the irregular meshes are the message-passing neural nets and the neural operator. These models could also be applied to particle-based simulations. It'd be great if the authors can include some of these models as baselines. 
  * Pfaff et.al; Learning Mesh-Based Simulation with Graph Networks
  * Brandstetter et.al; Message Passing Neural PDE Solvers.
  * Guibas et.al; Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers

2. Missing related works: I think there are some other associated works about symmetry and fluid dynamics that should also be discussed in the paper such as
* Brandstetter et.al; Lie Point Symmetry Data Augmentation for Neural PDE Solvers.
* Wang et.al; Incorporating Symmetry into Deep Dynamics Models for Improved Generalization.

3. It'd be great if the authors can also evaluate the models based on computational cost or inference speed. Because I think one of the biggest advantages of using neural nets for learning complex fluid dynamics is the computation efficiency.
questions: 1. Incorporating inductive biases into neural nets can make them more sample efficient. But I believe if there are enough training data, then the unconstrained networks can also learn the desired symmetry or conservation law from the data. Do you know how much data you need to make the unconstrained model perform as well as the constrained one? A plot of how the prediction performance of both constrained and unconstrained models change across different training set sizes would be helpful.

2. In Figure 6, there is no bar for the base model on the momentum evaluation metric. Does it mean the unconstrained model does not learn the conservation of momentum at all? That seems impossible.

3. Eqn 9: shouldn't it be the integration over the neighbor of x instead of the entire P_Q?
limitations:
review_title: null

Meta-review: metareview: The paper models fluid particle dynamics with continuous convolutions where the convolutional kernels in the final layer are constrained to be antisymmetric. This physical constraint enforces conservation of momentum.  Reviewers think this is a well-written paper. Related work suggested during the review period should be added.
confidence: Certain
Decision: Accept