Issues Regarding the Implementation and Claims in the Paper

Dear authors,

I have been working with the code provided in this repository and have some concerns related to the methods and claims presented in the associated paper. Here are the specific issues I have encountered:

Unrealistic Use of MSE in Fitness Function: The method described in the paper is designed to solve high-dimensional partial differential equations (PDEs), where the true solution to the PDE is inherently unknown. However, both the paper and the code use the mean squared error (MSE) between the true solution and the model-generated solution in the fitness function of the TLGP method. This relies on an unknown condition, which is fundamentally unreasonable.
Incorrect Claim Regarding Dimensional Consistency of Physical Laws: The paper states that "The physical laws are expressed in terms of mathematical equations, and these equations have the same form in all dimensions." ([Cao et al., 2024, p. 20642]). This statement contains a clear physical and mathematical fallacy. It violates physical and mathematical intuition, as solutions to PDEs do not generally behave the same along all axes and do not share the same form. The examples provided in the paper's experiments are special cases and not generally applicable. For example, the 2D Poisson equation $u{xx} + u{yy} = f(x, y)$, where $f(x, y) = e^y - \sin(x)$, has a source term that is not isotropic, and therefore the solution to the PDE is also anisotropic. As such, the method does not possess the universality claimed.
Limited Symbolic Solving Capability: The current implementation only supports a limited set of symbolic functions. For instance, in the example where the solution is $u = \sin(\pi x_1) (\pi x_2)$, the only symbols provided are $\sin$, $+$, and $*$. With such limited symbols, it is plausible that the correct solution could be obtained merely through combinatorial enumeration, which does not validate the effectiveness of the method. If the method allows for the specification of more symbols, please provide additional examples and code demonstrations. (As it stands, adding the statement pset.addPrimitive(math.cos, arity=1) at line 49 of Tr_GPSRPossion_sin.py results in a bug.)

I hope these points can be addressed and clarified. Thank you for your attention to these matters.

Best regards

Thank you very much for your interest in our work!

Q1: Unrealistic Use of MSE in Fitness Function A1: We merely sample a dataset from the true solution. In practical applications, this dataset can be obtained through sensor or by using traditional numerical methods like the Finite Element Method (FEM) to obatin high-precision approximate solutions at the sampled coordinates.

Q2: Incorrect Claim Regarding Dimensional Consistency of Physical Laws A2: Thank you for your thoughtful question. I understand your concerns about the statement that “physical laws are expressed in terms of mathematical equations, and these equations have the same form in all dimensions". It’s important to clarify that the “same form” refers to the underlying structure or pattern of the equations, not necessarily their exact mathematical expressions in different dimensions. For example, in the case of the Poisson equation you mentioned, $u{xx} + u{yy} = f(x, y)$, where $f(x, y) = e^y - \sin(x)$, the corresponding 1D Poisson equations would be $u{xx}= -\sin(x)$ and $u{yy} = e^y$. The “same form” here refers to the second-order partial derivatives of the unknown function, which is a common feature in both the 1D and 2D Poisson equations. I understand that our experiments have been primarily conducted on isotropic PDEs, which may have given the impression of a limitation in our approach. We acknowledge this limitation and appreciate your comment. In our future work, we plan to extend our experiments to anisotropic PDEs.

Q3: Limited Symbolic Solving Capability: A3: To add more symbols, you would need to modify the Poisson/parse_string.py file and ensure other parts of the code are adjusted accordingly to accommodate the new operator. For example, if you want to add the $\cos$, you would need to add a function similar to convert_power_sin(string) for the $\cos$ operator and use it accordingly.

Thank you for your response. However, I have some additional concerns.

A1: We merely sample a dataset from the true solution.

Q1: So, can I regard it as a typical symbolic regression method but with the physical regularization of a PDE introduced? Because it is not able to perform the task of solving a PDE independently; mathematically, the true solution of a PDE is inherently unobtainable. And if you have a dataset from the true solution, a symbolic regression method is enough to find the expression of it.

A2:The "same form" here refers to the second-order partial derivatives of the unknown function, which is a common feature in both the 1D and 2D Poisson equations.

Q2: This statement does not prove that the 1D equation is transferable to the 2D equation. Another simple example is the Poisson equation $$u{xx} + u{yy}= \sin(x^y)$$ with their source is inseparable. How can you easily transfer from another simple 1D Poisson equation?

A3: To add more symbols, you would need to modify the Poisson/parse_string.py file and ensure other parts of the code are adjusted accordingly to accommodate the new operator.

Q3: I think this operation is more complicated, and it makes the published work less easy to use and less meaningful. Because the PR-GPSR method can automatically support more symbols. I hope the author can add more symbols to the code published on GitHub later.

Q1: So, can I regard it as a typical symbolic regression method but with the physical regularization of a PDE introduced? Because it is not able to perform the task of solving a PDE independently; mathematically, the true solution of a PDE is inherently unobtainable. And if you have a dataset from the true solution, a symbolic regression method is enough to find the expression of it.

A1: If we have a dataset from the true solution, a symbolic regression method can indeed find an expression that fits this dataset. But it may not satisfy all the conditions and constraints of the original PDE. That’s why our method not only uses symbolic regression to fit the dataset but also takes into account the structure and physical constraints of the PDE to ensure that the generated solution not only fits the data but also satisfies all the conditions and constraints of the original PDE. This has been validated in one of our previous works.

Q2: This statement does not prove that the 1D equation is transferable to the 2D equation. Another simple example is the Poisson equation with their source is inseparable. How can you easily transfer from another simple 1D Poisson equation?

A2: In our work, we infer that the 1D equation is transferable to the 2D equation based on the observation that PDE described physical laws exhibit similar patterns or structures across different dimensions. We don't prove that the solution of 1D equation must be transferable to the solution of 2D equation. In the case of the Poisson equation you mentioned, I think we can infer the 2D Poisson equation by $u_{xx}=sin(x)$. [This needs to be verified experimentally: ) ]

Q3: I think this operation is more complicated, and it makes the published work less easy to use and less meaningful. Because the PR-GPSR method can automatically support more symbols. I hope the author can add more symbols to the code published on GitHub later.

A3: Thank you very much for your suggestion. This is a great suggestion! I will make modifications to the code to facilitate the addition of new operators when I have the time. Your feedback is greatly appreciated and is very helpful for our work.

grassdeerdeer / HD-TLGP

Issues Regarding the Implementation and Claims in the Paper #1