SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning

• The empirical evaluation of the proposal is conducted on either too simplistic but less effective preconditioners (SOR and SSOR), or a widely used preconditioner (AMG) with only one single parameter. It would be more informative if the authors experimented with more AMG parameters (such as the choice of the smoother and other coarsening parameters) and conducted a sensitivity analysis.

• The generation of training data is costly: (# data points) * (# grid searches) * (cost of one preconditioned solve)

• The training of symbolic regression can also be costly because of the sample efficiency of reinforcement learning.

• It needs to be clarified of the relationship between genetic programming in section 2.2 and the rest of the paper.

• Although the application of the proposed approach in this context may be novel, the reasoning behind why it could outperform a class of preconditioners based on graph neural networks (https://proceedings.mlr.press/v202/li23e.html, https://arxiv.org/abs/2405.15557) is not evident.
• The authors test their framework using three datasets (Biharmonic, Darcy Flow, and Elliptic PDE) with minimal variation in matrix size. It would be beneficial to validate their approach using the SuiteSparse Matrix Collection (https://sparse.tamu.edu/).
• I found some sections a bit challenging to follow. Could the authors consider reorganizing the paper or providing more detailed explanations for each step of SymMaP to enhance clarity?
• The values in the columns labeled "SymMap 1" and "SymMap 2" in tables 1, 2, 3 are not clear and would benefit from additional information to clarify the results.
• A comparison of the performance in a CPU environment using an MLP is not adequate to make a definitive assertion symbolic expressions possess equivalent expressive capabilities to neural networks in this scenario, effectively approximating the optimal parameter expressions.

The main weakness of this study is that it misses the crucial step of using the derived symbolic expression to generate the optimal preconditioner. I have carefully checked Figure 2 and do not find an explicit connection between the expression $\tau$ and the compiled preconditioner in the library. Some remarks are given in Section 5.3; however, the presented expressions depend on unclear variables $x_1, x_2, x_3$, so how to use them in preconditioner construction is unclear.

In addition, I can list the following weaknesses:

1. The manuscript does not explicitly present the parametrizations of considered PDEs and how these parameters are passed as input to RNN. Also, the authors ignore details on how training and testing sets are prepared.
2. I did not find the name of the linear solver used to evaluate the preconditioners, e.g., CG, GMRES, or smth else.
3. The incomplete Cholesky/LU preconditioner is not included in the comparison, although it is among the most powerful.
4. The authors do not report the runtime for training the presented pipeline (although for Darcy, the runtime is given in Table 6). Moreover, they do not discuss how many linear systems are needed to solve with the generated preconditioner to pay off the training costs compared to classical approaches like SSOR with the optimal parameter or ILU. Tables 1 and 2 show a gain in runtime, which is good, but how much time does the training of symbolic regression require?
5. For unknown reasons, the authors include a comparison with MLP. However, a more interesting comparison is replacing RNN with Transformer architecture and analyzing the results in the performance of linear solver and training runtime.
6. No theoretical guarantees on the performance of such an approach or motivation of the presented pipeline are presented, so the robustness of this approach remains unclear.