Advection equation on the domain of super ellipse

[kasahundirirsa]

Hello Dr. Lu Lu and Deepxde community.

I try to write a code for advection equation on super ellipse domain but, it is not work. How can write the domain? Please some body help me.

with bast regard.

import deepxde as dde
import matplotlib.pyplot as plt
import numpy as np
#from numpy import flot

def advection_eq_exact_solution(x, t):
"""Returns the exact solution for a given x and t (for sinusoidal initial conditions).

Parameters
----------
x : np.ndarray
t : np.ndarray
"""
return np.exp(-(n**2 * np.pi**2 * a * t) / (L**2)) * np.sin(n * np.pi * x / L)

def gen_exact_solution():
"""Generates exact solution for the advection equation for the given values of x and t."""
# Number of points in each dimension:
x_dim, t_dim = (256, 201)

# Bounds of 'x' and 't':
x_min, t_min = (0, 0.0)
x_max, t_max = (L, 1.0)

# Create tensors:
t = np.linspace(t_min, t_max, num=t_dim).reshape(t_dim, 1)
x = np.linspace(x_min, x_max, num=x_dim).reshape(x_dim, 1)
usol = np.zeros((x_dim, t_dim)).reshape(x_dim, t_dim)

# Obtain the value of the exact solution for each generated point:
for i in range(x_dim):
    for j in range(t_dim):
        usol[i][j] = advection_eq_exact_solution(x[i], t[j])

# Save solution:
np.savez("advection_eq_data", x=x, t=t, usol=usol)
# Load solution:
data = np.load("advection_eq_data.npz")

def gen_testdata():
"""Import and preprocess the dataset with the exact solution."""
# Load the data:
data = np.load("advection_eq_data.npz")
# Obtain the values for t, x, and the excat solution:
t, x, exact = data["t"], data["x"], data["usol"].T
# Process the data and flatten it out (like labels and features):
xx, tt = np.meshgrid(x, t)
X = np.vstack((np.ravel(xx), np.ravel(tt))).T
y = exact.flatten()[:, None]
return X, y

Problem parameters:

a = 0.4 # Thermal diffusivity
b = 1
c = 1
L = 1 # Length of the bar
n = 1 / 2 # Frequency of the sinusoidal initial conditions

Generate a dataset with the exact solution (if you dont have one):

gen_exact_solution()

def pde(x, y):
"""Expresses the PDE residual of the advection equation."""
dy_tt = dde.grad.jacobian(y, x, i=0, j=1)
dy_xx = dde.grad.hessian(y, x, i=0, j=0)
return dy_tt + a * dy_xx

Computational geometry:

geom = dde.geometry.Interval(0, L)
timedomain = dde.geometry.TimeDomain(0, 2 * np.pi)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

Initial and boundary conditions:

bc = dde.icbc.DirichletBC(geomtime, lambda x: (abs(b) * ((np.cos(x)) ** 2 / n)), lambda t: (abs(c) * ((np.sin(t)) ** 2 / n)))
ic = dde.icbc.IC(
geomtime,
lambda x: np.sin(n * np.pi * x[:, 0:1] / L),
lambda _, on_initial: on_initial,
)

Define the PDE problem and configurations of the network:

data = dde.data.TimePDE(
geomtime,
pde,
[bc, ic],
num_domain=2540,
num_boundary=80,
num_initial=160,
num_test=2540,
)
net = dde.nn.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
model = dde.Model(data, net)

Build and train the model:

model.compile("adam", lr=1e-3)
model.train(epochs=200)
model.compile("L-BFGS")
losshistory, train_state = model.train()

Plot/print the results

dde.saveplot(losshistory, train_state, issave=True, isplot=True)
X, y_true = gen_testdata()
y_pred = model.predict(X)
f = model.predict(X, operator=pde)
print("Mean residual:", np.mean(np.absolute(f)))
print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))

[kasahundirirsa]

Dr. Lu Lu
The problem is on the boundary condition. How can rewrite it?

[lululxvi]

Could you add more details?

[kasahundirirsa]

I try to solve advection equation using ANN on the domain of super ellipse.

[kasahundirirsa]

Dr. Lu Lu
This is the error on my code

TypeError: () takes 1 positional argument but 2 were given

[kasahundirirsa]

I try to solve advection equation using ANN on the domain of super ellipse.

[kasahundirirsa]

'''''''''''''''''''''''This is the domain what i need. ''''''''''''''''

import matplotlib.pyplot as plt
import numpy as np
a = 1
b = 1
n = 0.5
t = np.linspace(0, 2 * np.pi, 1000)
x = ((np.abs(np.cos(t))) ** (2 / n)) * a * np.sign(np.cos(t))
y = ((np.abs(np.sin(t))) ** (2 / n)) * b * np.sign(np.sin(t))

plotting the curve

plt.axis('equal')
plt.plot(x, y)
plt.colorbar()
plt.show()

[lululxvi]

I didn't get your question. What is the difficulty? What have you tried?

[kasahundirirsa]

Dr. Lu Lu Ok, as i told you by using Deepxde library, try to code using ANN for solving advection equations on super ellipse geometry. So the result must be on super ellipse geometry. Q 1, How to get the super ellipse result ? [image: supar ellipse.png] The result on the above graph for the advection equation. Please can I get the code?

…

On Wed, Jun 15, 2022 at 11:36 PM Lu Lu ***@***.***> wrote: I didn't get your question. What is the difficulty? What have you tried? — Reply to this email directly, view it on GitHub <#732 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AYV4JA6OW4X5K4TNI4NAYRLVPI5FJANCNFSM5YN2IMSA> . You are receiving this because you authored the thread.Message ID: ***@***.***>

[lululxvi]

DeepXDE doesn't support super ellipse geometry. You need to either implement a new geometry or use anchors. See FAQ Q: More details and examples about geometry. and Q: How does DeepXDE choose the training points? How can I use some specific training points?

[kasahundirirsa]

Ok, thanks. What about for circles in 2d advection equation.

…

On Wed, 22 Jun 2022, 7:20 pm Lu Lu, ***@***.***> wrote: DeepXDE doesn't support super ellipse geometry. You need to either implement a new geometry or use anchors. See FAQ Q: More details and examples about geometry. and Q: How does DeepXDE choose the training points? How can I use some specific training points? — Reply to this email directly, view it on GitHub <#732 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AYV4JAY7SRMJPERR3M2B7JTVQM4NJANCNFSM5YN2IMSA> . You are receiving this because you authored the thread.Message ID: ***@***.***>

[lululxvi]

Circle is supported, see https://deepxde.readthedocs.io/en/latest/modules/deepxde.geometry.html

[kasahundirirsa]

Thanks, I'll check it out.

…

On Mon, Jun 27, 2022 at 9:18 PM Lu Lu ***@***.***> wrote: Circle is supported, see https://deepxde.readthedocs.io/en/latest/modules/deepxde.geometry.html — Reply to this email directly, view it on GitHub <#732 (reply in thread)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AYV4JA3JX76465WTED3RXFDVRHV5TANCNFSM5YN2IMSA> . You are receiving this because you authored the thread.Message ID: ***@***.***>