Kati8/ICLR_2023_2025 huggingface dataset:
authors: Chonghan Gao
authors: Haoyi Zhou
authors: wen xin gong
authors: QING PO WU WU
authors: Tianyu Chen
authors: Qian Yu
authors: Jianxin Li
reviews: 
Comments: This paper proposes a diffusion-based model, called PDE-Diffusion, to solve partial differential equations. This method is a two-stage model consisting of an autoencoder and a diffusion model. PDE-Diffusion incorporates physics-based priors to enhance model interpretability and generalization. Moreover, the model assimilates PDE-informed constraints to ensure temporal coherence while producing high quality predictions.
Strengths: - The proposed method exhibits better interpretability and generalizability by embedding physical constraints into the reverse process of diffusion model. 
- The method adopts a latent residual to mitigate the problem of temporal incoherence in physical field predictions.
- This paper introduce two new datasets that could enhance the available resources for using deep learning to solve PDEs.
Weaknesses: - There’s other diffusion-based PDE solver that should be compared to, such as DYffusion [1]
- The baseline is not enough. FNO is not a recent method and not the state-of-the-art model. I recommend authors to compare with more recent neural PDE solvers to better evaluate the effectiveness of the proposed method, such as FFNO [2] and GFNO [3]. 
- In most rows in table 2 and 3, the MSE and MAE of PDE-Diffusion is larger than the other baselines but is still annotated boldface, which is misleading. 
- In section 5.3, many placeholders are not replaced by final results.


[1] Cachay, Salva Rühling, et al. "DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting." arXiv preprint arXiv:2306.01984 (2023).\
[2] Tran, Alasdair, et al. "Factorized fourier neural operators." arXiv preprint arXiv:2111.13802 (2021). \
[3] Helwig, Jacob, et al. "Group Equivariant Fourier Neural Operators for Partial Differential Equations." arXiv preprint arXiv:2306.05697 (2023).
Recommendation: 3: reject, not good enough
Reviewer confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness/correctness: 2 fair
Clarity: 2 fair
reviews: 
Comments: To address the challenge of interpretability, generalizability and long-horizon predictive performance, the author proposed the PDE-Diffusion model. The model incorporate physics-based priors, two stage diffusion model to tackle the mentioned challenges. It was also tested on extensive datasets.
Strengths: Originality: The author propose to embed a physics-based prior into the sampling process for diffusion model to enhance the learning result. 

Quality and clarify: The methodology part misses details about the physics-based priors. The result part is confusing, indicating the proposed model cannot beat the SOTA models.

Significance: The author claims that the physics-embedded framework is more interoperable and generalizable, meanwhile mitigate the problem of temporal coherence.
Weaknesses: The methodology part is not clear in cases of how to embed the physics prior $\epsilon_I$ and $\epsilon_B$ during the reverse sampling process. The result doens't support the claim where in many cases, the PDE-Diffusion result is much worse than FNO. The reviewer also has major concern about the writing style. It seems not rigorous and scientific on delivering the message. Details can be found in the questions part.
Recommendation: 3: reject, not good enough
Reviewer confidence: 5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.
Soundness/correctness: 3 good
Clarity: 1 poor
reviews: 
Comments: The paper considers a two-stage diffusion-based model for solving partial differential equations (PDE). It conducts experiments on 4 benchmark datasets to compare its performance with some leading methods.
Strengths: The paper explores a diffusion-based method for solving partial differential equations. It compares the diffusion-based method performance with some leading methods on four benchmark datasets.
Weaknesses: 1. The manuscript is essentially unfinished. There are many typos, missing figures in the table (e.g. Table 2. FNO MAE CFD2D turb2), missing values (e.g. 5.3 Results Analysis: The MSE decrease is x% (at 2), x% (at 4)...). 

2. The comparative results show that FNO is better than the proposed. Table 3 shows that except for one or two cases, the PEDM performs worse compared to FNO. In Table 3, the average of the six random initializations with variance should have been reported rather than the individual results of the initialization. 

3. The paper fails to compare with more advanced techniques than FNO.

4. The motivation for the use of the diffusion model for solving PDE is not sound. Diffusion is founded on a probabilistic setting while PDE is deterministic and it isn't easy to relate the two.

5. The inference time comparison between the different methods should be shown in comparison. FNO finds the solution in a parallel manner while PEDM does not.
Recommendation: 1: strong reject
Reviewer confidence: 5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.
Soundness/correctness: 1 poor
Clarity: 1 poor