Summary
The work proposes a generic method to incorporate Lie point symmetry into physics-informed neural networks (PINNs) by augmenting loss function. The method leverages automatic differentiation as other PINNs do because the condition for symmetries is written using differentials. The authors demonstrated the model with symmetry gained better performance than one without symmetry.
Strengths
* The work is based on an established mathematical theory found in Olver (1986).
* Once the Lie point symmetries are obtained, it seems quite simple to incorporate them, thanks to automatic differentiation and the PINNs framework.
Weaknesses
* The experiments look insufficient. They compared with and without symmetry but no other state-of-the-art PINN-based models. It makes the contribution of the work look limited because most of the mathematical contributions directly come from Olver (1986), and the results from the symmetry model are not good enough (in particular, Figure 2). They claimed that "our goal is to showcase the effectiveness of using symmetries" (lines 230-231). However, the positive effect of the symmetries is well-known to the community (e.g., Wang et al. 2021a).
* The superiority of incorporating symmetries through loss function is unclear. I suggest the authors include data augmentation methods (e.g., Brandstetter et al. 2022a) and equivariant models (e.g., Wang et al. 2021a).
* The authors claimed, "Our work presents the foundations for leveraging Lie point symmetry in a large family of Neural PDE solvers that do not require access to accurate simulations" (lines 316-318). However, another part says, "The data for Burgers' equation is obtained using the Fourier Spectral method" (line 299). It makes the reviewer confused about whether the method uses simulation data as supervisory signals.
* The contribution of the paper is not clear enough. Is it a new theorem proved, an undiscovered observation found, a state-of-the-art performance, or a novel problem setting? The reviewer recommends the authors clearly state the contribution of the work and show the supporting facts.
* The mathematical presentation lacks correctness and clarity. The reviewer found no clear definition of "Lie point symmetry." The definition of $u^{(n)}$ lacks $u$ (see Olver (1986) p.97). It may be by mistake; however, this is an essential part of the theory. Thus the reviewer recommends the authors carefully re-check the manuscript for more mathematical correctness. For instance, if there is no $u$ in $u^{(n)}$, one cannot express the advection term of the Burgers' equation in the jet space.
Minor points:
* Please add an explanation about $N_s$.
* The paper uses "viscosity" for the heat equation, which seems uncommon. It could be called the diffusion coefficient.
* In the first equation of Equation (1), $t$ should be defined in an open set (see, e.g., Jürgen Jost "Partial Differential Equations ThirdEdition" (2012) Chapter 1). That's why we need the initial condition.
Questions
* How long do the training and prediction take, respectively?
* What is the meaning of "orthogonality" in the paper, e.g. "our symmetry loss encourages the orthogonality of $\mathrm{pr}^{(n)}v$ and the gradient of $\Delta$" (line 237)? The condition contains PDE itself, so the reviewer guesses the use of the terminology could require some explanation.
Rating
3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.
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.
Limitations
The limitations are stated sufficiently in the paper.