Helmholtz PDE

[alabaykazakh]

Dear Hongkai,

Thank you for your very interesting research. Last half a year, I was aiming to solve the Helmholtz equation with PINNs using DeepXDE packages. I could reach a mean average percentage error of 0.06% for some cases. Now I am looking for a proper approach to solve the parametric Helmholtz PDE with up to 20 extra input parameters besides the spatial coordinates to develop a generalized PINN. From experiments, PINN itself handled parametric PDE with worse accuracy, so this makes me have a look at PINO.
Q1: Transferring PINN to PINO is quite painful for me. Do you have a successful experience with training PINN on multidimensional input? At the moment, it looks like all my attempts are in vain, so do you know if PINO/FNO/DeepONet is the better way to solve parametric PDEs?
Q2: My boundary conditions are hard-constrained periodic ones, so no exact values are known like in the case of Burger's equation. I cannot do purely-physics-informed training in such a case and have to feed the operator with target data?

[zongyi-li]

Thank you for your interest in our works!
A1: We also had a similar experience training PINNs on multidimensional input. It usually requires very careful design for the branch net. On the other hand, the operator based methods may have an advantage. They (1) can takes the advantages of the existing training data (from similar cases of coefficients/boundary etc) and (2) they often have a better optimization landscape. Usually the operator based methods can get a decent error rate for the Helmholtz equation (as long as the frequency is not too high).
A2: the PINO uses fourier representation and they are designed best for problems with periodic boundary. It seems your problem will be a perfect case. When using fourier representation, the periodicity is assumed so you don't have to impose a boundary loss.