reviews: 
Comments: This paper aims to leverage a diffusion-based model for solving Partial Differential Equations (PDEs), targeting improvements in generalizability, interpretability, and long-term predictive performance. Although the proposed PDE_Diffusion model exhibits some improvement in error metrics compared to other deep learning-based approaches, the paper lacks a rigorous justification for why diffusion models are well-suited for solving deterministic PDEs and how it should work.

I think the paper suffers from various technical and presentation issues. I recommend rejecting the paper for the following reasons: (1) the methodology is poorly elucidated, accompanied by ambiguous notations and questionable mathematical arguments, and (2) it fails to provide a compelling rationale for employing diffusion models, which are inherently probabilistic, in solving deterministic PDEs.
Strengths: The paper demonstrates through experiments that diffusion models may hold promise for solving PDEs.
Weaknesses: Major Comments:

1. Diffusion models are conventionally used to model distributions, focusing on generating samples that are similar to the training dataset. In contrast, this paper focuses on a deterministic task: solving a well-defined PDE. How do diffusion models contribute to solving such deterministic problems? Is the sample generated by the reverse SDE consistently close to the deterministic ground truth with small variance? If so, can one replace the reverse SDE with a deterministic process and initial conditions to obtain deterministic and improved results?

2. Although I am familiar with PDEs and diffusion models, I find the methodology section difficult to follow due to unclear notation and inexact mathematical expressions. There are many problems, I just list a few.

  (a) In the first paragraph of section 4.2, $\mathbf{D}$ denotes a decoder. According to $\mathbf{D}(z^0, d^i)$, $d^i$ should be a single snapshot in the latent space as residual. However, the next line states $d^i = (d^1_1, \dots, d^i_N)$, consisting of all the intermediate steps in the diffusion process.

  (b) What is the meaning of "probability that $d_m^i$ conforms to the governing equations $e$"?

  (c) Equations (5)(6)(7)(8) appear to model a conditional diffusion model. However, it is known in the literature that there is no straightforward way to model a conditional diffusion model directly from the original diffusion model without certain approximations (see e.g., the discussion in (Song et al., 2020b)). Since the paper does not clarify the meaning of conditional on $e$, it is unclear what this section is attempting to convey. Equation (6) seems more like an assumption than a mathematical derivation.

  (d) Equation (6) involves the notation $\hat{u}^{i-1}_n$ and $\hat{u}^{i}_n$, but the next line discusses the notation $\hat{u}^{n}_i$ and $\hat{u}^{n}_{i-1}$.

  (e) What is meant by "derivative kernel" above equation (12)?

3. The paper claims that solving some PDEs takes "months-long simulations utilizing iterative numerical solvers." However, I believe the PDEs tested in this paper are far from such a regime. Solving these PDEs should be relatively quick, perhaps just a few minutes. In contrast, each physical-time evolution step in the proposed PDE-Diffusion model involves multiple diffusion steps, which can be time-consuming, not to mention the extensive training time required for these diffusion models. The authors should provide the solving time of classical solvers, inference time per solution, and total training time for diffusion models to give readers a sense of the time cost.

4. There are placeholders in the paper where results should be reported. For instance, on page 7, it states "PDE-Diffusion has a decrease of x% on average in the PRE metric" and "Our method shows significantly better results in the case of longer prediction window sizes. The MSE decrease is x% (at 2), x% (at 4), x% (at 8), x% (at 16)." These "x%" placeholders indicate hasty composition, suggesting the authors did not even thoroughly proofread the paper.

5. The paper cites (Gao et al., 2022) when introducing the significance of solving PDEs in the first sentence. However, this paper has a very loose connection with PDEs. A keyword search for "PDE" in that paper reveals that the term only appears twice in the related work section. Citing such a work (Gao et al., 2022) with minimal relevance to the paper's focus on PDEs, particularly in the opening sentence, is quite unprofessional.

Minor Issues:

1. What is the "PRE metric"? The paper lacks a definition.

2. In the problem definition, the authors present solving a PDE as finding an approximate solution on a fixed spatial discretization. This is misleading. In the field of numerical PDEs, the choice of spatial discretization is integral to the PDE-solving process, and most numerical methods can accommodate varying discretizations. The authors should clarify that their method is not mesh-free.

3. Typo in the appendix: "which is reasonable when reasonable in the limit of infinite diffusion steps".

4. In several instances, the authors refer to "partial derivative equations." This term is unconventional; Google search results suggest it is rarely used.
Recommendation: 3: reject, not good enough
Reviewer confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Soundness/correctness: 1 poor
Clarity: 2 fair