Venue: NeurIPS 2022 Conference

Reviews:

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 of this paper propose a physics based generative model named Possion flow generative model (PFGM). In this model, the source function corresponds to the data distribution; and a dual ODEs serve as mapping between the data distribution and the positive hemisphere with radius r. The generative process starts from a point on the hemisphere, follows the negative electric field, and finally generate a sample. This model was tested on CIFAR-10 and CelebA and compared with state of the art image generation models including score-based methods and obtained comparable or better performance. Furthermore, it is shown that PFGM's generation process is much faster than existing methods.
strengths_and_weaknesses: Strengths:
1. Inspired by Poisson equation in physics, the idea of this deep generative model is interesting. To avoid generated data points collapse at one point, augmented dimension is applied.  
2. The generation process in PFGM is more efficient than existing models for image generation. 

Weaknesses:
1. Quantitative analysis unveils that PFGM seems to work better on larger datasets. This should be furthered investigated on a third dataset. 
2. The authors should discuss and compare (if possible) with ViT based generative model, e.g., https://arxiv.org/abs/2107.04589. 
questions: 1. It is highly recommended that PFGM should be investigated on a third dataset.
2. It would be interesting to see a direct comparison with ViT based image generation models. 
limitations: Potential social impacts are discussed in the supplementary file.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: The manuscript proposes using the Poisson equation in higher dimension to use as a generative model. Several numerical case studies are reported to illustrate the benefit of the proposed model in terms of computational speed and accuracy.
strengths_and_weaknesses: - The paper proposes an interesting physics-motivated heuristic for generative models. Despite the initial idea being interesting, the reviewer gets the feeling that the authors did not spend efforts to make the ideas rigorous instead focused more on showcasing the idea on some examples through more fixes and heuristics along the way.

- Typos and carelessness galore. Some examples: $G(\boldsymbol{x},\boldsymbol{y})$ in equation (2) is valid only if the dimension $N\geq 3$, otherwise (for $N=2$) the formula is not valid and we need a logarithmic fundamental solution. The manuscript did not state this assumption on $N$. Line 191: "radical projection" --> "radial projection". In line 86: the positive function $f(\boldsymbol{x})$ is introduced without any other restrictions. We cannot guarantee Lipschitzness of the vector field $f(\boldsymbol{x})\boldsymbol{E}(\boldsymbol{x})$, and hence the existence-uniqueness of the forward and backward flows without any assumptions whatsoever. Notice that here regularity of $\boldsymbol{E}$ inherited from $\varphi$ which is in turn inherited from the regualrity of $\rho$, is not enough for the forward and backward ODE solutions' existence-uniqueness. In line 108: $p(\boldsymbol{x})$ is introduced as a probability distribution while in line 109 it is equated with $\rho(\boldsymbol{x})$ which was defined to be a density in line 68.

- The proofs, like the writing throughput the paper, are all heuristic lacking mathematical rigor and ignores all technical issues similar to the above. For instance, the crucial issue of whether the Poisson equation is guaranteed to have unique $C^2$ solution is not treated. 
questions: - What is the mathematically rigorous justification of making the time derivative of $\boldsymbol{x}$  proportional to $\boldsymbol{E}(\boldsymbol{x})$? Can you do away with the heuristic argument in lines (84)-(85) and give mathematical arguments? Otherwise, one can think of infinitely many generative models (why just hemisphere?) and infinitely many vector fields for the same geometry of the Poisson equation.


limitations: The authors have stated some limitations. However, the main limitation I see, that is, the mathematically rigorous motivation/justification of the proposed method, is not discussed. I would have expected the authors to acknowledge this and put it in the Section I.
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: This paper propose a generative model based on Poisson equation. The authors augment each data point with a scalar variable z and model the true data point as points centered as $z = 0$  hyperplane. During training, the authors define a empirical Poisson field based on the current batch of data and run the forward ODE to calculate the normalized field $V_{B_L}$. They train the model $f_\theta$ to regress on $V_{B_L}$ via L2 loss. During the backward process, the author start from a large $z = z_{max}$ and graduately walk back to $z=0$ to generate samples. Using this model, the authors try to solve the problem in the original VE/VP ODES that the $l2$ norm of perturbed training data and the standard variance of Gaussain perturbation kernel may have strong correlation. They use various experiments to demonstrate the performance and effectiveness of the propose model.
strengths_and_weaknesses: For the strength of the paper:

I think this model proposed a new form of the SDE/ODE based generative model. By using Poisson's equation, the proposed PFGM model may have weaker $norm-z(t')$ correlation and propose good samples with less computation. The paper is well written and the ideas are easy to follow.  

For the weakness:
1. The first weakness of the proposed method is the large computation needed during training. As shown and discussed in Limitations and Future Directions in the appendix, using more complex Poisson field, the PFGM is more than 2 times slower than the score-based models. This may limit the model from going to high-resolution and high-quality data. 

2. Also as pointed out by the authors, besides the training time, the model defines an empirical Poisson field based only on a subset of the data, which means at each step, the  model $f_\theta$ is trying to fit a different underlying distribution and the estimation is biased. Although the authors try to mitigate the problem by using larger batch size, as the image space is highly multimodal, just enlarging batch size is not the final solution. This might also prevent the model from scaling up.
3. I also have some questions, please check the next part.
questions: 1. The sample quality: The authors claim that their model achieves comparable and even better performance as [1, 2] based on the scores reported in Table 1 and 2. However, the samples shown in Figure 3 seems to be worse than that shown in [1]. For example, comparing Figure 3 with Figure 14 in [1], it seems that the face samples in Figure3b contain more obvious artifacts. Do you have any explanation for that?
Also, the current paper only shows 64 x 64 samples while [1,2] can do 256 x 256. 
Thus, it might be hard to judge whether the proposed method does improve the baseline.

2. The sampling time: Table1 and Table2 compare the sampling time through NFE. But since PFGM uses different ODE equation, would it be any time difference for runing each ODE step? Or would you please also comparing the total sampling time (maybe based on wall-clock time)?

3. The sensitivity of the hyperparameters: The proposed model introduce several hyperparametes like MAX_STEP, $z_{min}$, $z_{max}$, etc. How sensitive is the model if we change these parameters. Especially, Theorem 1 assumes $r \rightarrow \infty$. Then do we need to  carefully choose $z_{max}$ to meet this requirement?

[1] Improved techniques for training score-based generative models.

[2] Score-Based Generative Modeling through Stochastic Differential Equations 

limitations: The authors have adequately addressed the limitations and potential negative societal impact of their work.
review_title: null

Meta-review: metareview:  All the reviewers agreed that the paper is novel and interesting with significant contributions. While there were certain concerns regarding the experimentation, clarity, and mathematical rigor initially, the extensive rebuttal provided by the authors addressed most of the concerns, hence some reviewers increased their scores. Hence, I am happy to recommend an acceptance for the paper.

However, I must say that, I find the phrase "record breaking" academically inappropriate and I kindly request the authors to replace it with a more academic phrase, such as achieving the state-of-the-art. 
confidence: Certain
Decision: Accept