DeepFDM: A scientific computing method for Neural Partial Differential Equation (PDE) operators

• The motivation of choosing the known family of time dependent PDEs with periodic boundary conditions, and bounded coefficients is not clear to me. The problem setup seems very restricted, and the method is only applicable to learn from initial conditions.
• It should be made clear that if the method is data-driven/known PDE, PDE solver/operator learning in the beginning. According to my understanding, the PDE needs to be known to use the finite differences solver. DeepFDM learns an operator from the initial condition for a specific family of PDEs to the solution at next time step, and the iteratively solve for a longer time. The problem setup should be more rigorous.
• The literature review in Section 2.1 is not well organized or well-written. The papers of PDE discovery, PINN and operator learning are mentioned without a focus. Some claims are not correct and language is vague. For example, “Lu et al. (2019) propose the DeepONet architecture, which learns PDE solution operators. However, in this case, the PDE is fully known and the PDE residual is included in the loss.” It is not correct. There is no PDE known in vanilla data-driven DeepONet. The authors may refer to Physics-informed DeepONet. “Neural PDE operators aim to learn to solve a given PDE from data, without assuming that the form of the PDE is known.” This claim is conflicting with the above point.
• One main issue is that it ‘s not fair to compare DeepFDM with FNO, U-Net, and ResNet, since the PDE structure is known and of course it can perform better than pure data-driven methods. This makes the results not convincing.
• There is an existing paper on distribution shift quantification: M. Zhu, H. Zhang, A. Jiao, G. E. Karniadakis, & L. Lu. Reliable extrapolation of deep neural operators informed by physics or sparse observations. Computer Methods in Applied Mechanics and Engineering, 412, 116064, 2023.
• Some notations are clearly defined. For example, m in the dataset, A, A* and \hat{A}.
• There are no metrics for coefficient fields if the author considers solving the inverse problem.
• I don't see Appendix B in the manuscript.
• "In this case, the solution is generated on a higher resolution grid, and then coarsened (upsampled)." It should be "downsampled".

1. The paper is not very clear in the type of problem it approaches. The writing could be improved to make the definition of the problem more easy to understand.
2. The usual setup in neural PDE solvers is that the PDE parameters are known. In this setting, neural PDE solvers have already been compared to numerical methods. The authors could better motivate their specific choice of problem definition (i.e., unknown, spatially-dependent PDE parameters).
3. The method is only a useful baseline if the neural PDE solver is trained on real-world data. When the neural PDE solver is used as a surrogate for a numerical solver, the PDE parameters would be known (since they would have been used to generate the training data).
4. There is no inference time evaluation. Faster inference is one of the main reasons for utilizing a neural PDE surrogate instead of a numerical method like the one considered in the paper.
5. Many experimental and model details are missing (see questions).

DeepFDM is presented as a benchmark method; however, its applicability is limited to a specific class of partial differential equations (PDEs). The paper does not sufficiently discuss how this restriction affects DeepFDM's generalizability, particularly in scenarios that require flexibility across various forms of PDEs, such as nonlinear PDEs and complex boundary conditions.

For instance, in the case of hyperbolic equations with shock locations, finite difference methods (FDM) may struggle to accurately capture the discontinuities inherent in these solutions. This limitation could significantly impact the performance and reliability of DeepFDM when applied to a broader range of PDE types.

Please explicitly state the objectives and justify the choice of comparison methods in the context of those objectives. More specifically, Why do the authors compare DeepFDM to both neural networks like ResNet and Unet, as well as neural operators like FNO? It’s unclear whether the authors aim to solve individual instances of PDEs or to learn a solution operator.

1. The main weakness of this work is that the authors combined two different contributions in a single study: a dataset generation procedure for benchmarking neural PDE solvers and a DeepFDM method for fitting the PDE coefficients.
2. The idea of parameterizing the finite-difference method via CNN is not new and has already appeared in other works like the smoothing operator in the multigrid methodб https://arxiv.org/abs/2102.12071
3. The proposed method's scalability is not discussed or compared with competitors.
4. The presentation of the problem statement is confusing since the authors start not from the inverse problem of coefficient reconstruction but from the solution reconstruction problem.