studyhardhardhard
commented
2 months ago Dear Developer,
I hope this message finds you well. I understand you must be very busy, which is why I haven't received a reply from you for over a month. I apologize for reaching out again. Recently, I had the pleasure of reading your new work, “An Analysis and Solution of Ill-Conditioning in Physics-Informed Neural Networks,” and I was delighted to see the further development of TSONN.
Since I didn’t receive a response from you, I took the time to further explore TSONN, particularly focusing on the problem of solving Laplace's equation for a uniform flow coupled with dipoles. In this problem, a uniform flow is applied at the far field, while near the wall, only natural boundary conditions are imposed. This implies that the partial differential equation involves only Neumann boundary conditions without any Dirichlet boundary conditions, potentially leading to multiple solutions that differ by a constant C.
In the absence of specific code references, I tested my hypothesis using a vanilla PINN. In this scenario, even if only a single point is constrained to determine the constant C, the equation yields a unique correct solution rather than multiple solutions. Even a vanilla PINN can solve the exact same Laplace problem you described.
I am also very curious about the remarkable effectiveness of your TSONN, which seems capable of resolving the issue of multiple solutions with only Neumann boundary conditions. However, without direct implementation details, further analysis seems challenging.
Nevertheless, I remain highly interested in TSONN and wish you continued success in your research.
Best regards.
Dear Developer,
I have thoroughly read your paper "TSONN: Time-stepping-oriented neural network for solving partial differential equations," and I find it to be outstanding with significant practical value. Thank you for sharing your code! However, it seems that section 3.1, concerning the two-dimensional Laplace’s equation for solving irrotational, incompressible flow past a cylinder, is missing the relevant source code. I wonder if this was an oversight? If you could make all the source code for TSONN available, it would likely be beneficial to me and others. Of course, this decision is entirely up to you, but I believe that TSONN has substantial potential, and providing more comprehensive results could be advantageous for a wider audience.
Additionally, I have a small question. Since Laplace’s equation is time-independent and linear, why is it that PINN cannot solve it effectively? I might be missing something, but I haven't seen many examples in the literature of PINN being used to solve Laplace’s equation. If the time-independent Laplace equation cannot be well-solved by PINN, why can it be addressed through time-stepping? Given my limited knowledge, any guidance would be highly appreciated.
I wish you continued success in your research on TSONN.
Best regards.