How to implement boundary conditions on Helmholtz equation on a disk?

[Shnrqpdr]

Hello everyone.

In my study of PINNs, i want to implement a Helmholtz equation on a disk with center in 0 and radius 1. The solution involves a superposition of Bessel functions and sin and cos functions. I want to implement a Dirichlet boundary condition for the function that vanishes at r=1.

In other words, I want to apply PINN in the solution of a vibrating membrane.

Here is my code

import deepxde as dde
import numpy as np
# Import tf if using backend tensorflow.compat.v1 or tensorflow
from deepxde.backend import tf
import scipy
import numpy as np
import scipy.special as sc
import pandas as pd

A = 5.0
B = 5.0
C = 5.0

def pde(x, y):
    k0 = 1.0
    dy_r = dde.grad.jacobian(y, x, i=0, j=0)
    dy_rr = dde.grad.hessian(y, x, i=0, j=0)
    dy_thetatheta = dde.grad.hessian(y, x, i=1, j=1)

    return x[:, 0:1] * dy_r + x[:, 0:1] ** 2 * dy_rr + dy_thetatheta + k0**2 * x[:, 0:1]**2 * y 


def solution(x):
    r, theta = x[:, 0:1], x[:, 1:]
    sum = 0.0

    for i in range(10):
        zeros_j = sc.jn_zeros(i, 3)
        for zero in zeros_j:
            sum = sum + (C*sc.jn(i, r*zero)*(A*np.cos(i*theta) + B*np.sin(i*theta)))

    return sum

geom = dde.geometry.Rectangle(xmin=[0, 0], xmax=[1, 2 * np.pi])
bc_rad = dde.icbc.DirichletBC(
    geom,
    lambda x: 0,
    lambda x, on_boundary: on_boundary and dde.utils.isclose(x[0], 1),
)

def feature_transform(x):
    return tf.concat(
        [x[:, 0:1] * tf.sin(x[:, 1:2]), x[:, 0:1] * tf.cos(x[:, 1:2])], axis=1
    )

data = dde.data.PDE(
    geom, pde, bc_rad, num_domain=2540, num_boundary=80, solution=solution
)

net = dde.nn.FNN([2] + [80] * 2 + [1], "tanh", "Glorot normal")
net.apply_feature_transform(feature_transform)

#net.apply_output_transform(transform)

model = dde.Model(data, net)
model.compile("adam", lr=1e-3, metrics=["l2 relative error"])
losshistory, train_state = model.train(iterations=15000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)

But the solution that the NN provides is far away of the correct solution.

So, here is my question:

How to implement a dirichlet boundary conditions on a helmholtz equation on a disk that vanishes at r=0 ? Why and where my implementation is wrong?

Best regards,

[lululxvi]

Use polar coordinates?