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poisson_2d.py
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poisson_2d.py
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from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import numpy as np
import deepxde as dde
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import make_axes_locatable
from deepxde.backend import tf
# PINN
a = 10
def f(x, y):
u_xx = (
16**a
* a
* (a * (1 - 2 * x) ** 2 - 2 * x**2 + 2 * x - 1)
* ((x - 1) * x * (y - 1) * y) ** a
/ ((x - 1) ** 2 * x**2)
)
u_yy = (
16**a
* a
* (a * (1 - 2 * y) ** 2 - 2 * y**2 + 2 * y - 1)
* ((x - 1) * x * (y - 1) * y) ** a
/ ((y - 1) ** 2 * y**2)
)
return -u_xx - u_yy
def output_transform(x, y):
x_in = x[:, 0:1]
y_in = x[:, 1:2]
return x_in * y_in * (1 - x_in) * (1 - y_in) * y
def pde(x, y):
x_in = x[:, 0:1]
y_in = x[:, 1:2]
du_xx = dde.grad.hessian(y, x, i=0, j=0)
du_yy = dde.grad.hessian(y, x, i=1, j=1)
return du_xx + du_yy + f(x_in, y_in)
geom = dde.geometry.Rectangle([0, 0], [1, 1])
data = dde.data.PDE(geom, pde, [], num_domain=400)
net = dde.maps.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
net.apply_output_transform(output_transform)
PINNmodel = dde.Model(data, net)
PINNmodel.compile("adam", lr=1.0e-3)
losshistory, train_state = PINNmodel.train(epochs=20000, callbacks=[])
def gen_test_x(num):
x = np.linspace(0, 1, num)
y = np.linspace(0, 1, num)
l = []
for i in range(len(y)):
for j in range(len(x)):
l.append([x[j], y[i]])
return np.array(l)
def sol(t):
x = t[:, 0:1]
y = t[:, 1:2]
return (16 * x * y * (1 - x) * (1 - y)) ** a
x = gen_test_x(100)
print(
"L2 relative error of u",
dde.metrics.l2_relative_error(sol(x), PINNmodel.predict(x)),
)
##########################################
plt.rcParams.update({"font.size": 22})
plt.figure()
X = gen_test_x(100)
y = PINNmodel.predict(x).tolist()
disp = []
prev = X[0][1]
temp = []
for i in range(len(y)):
if X[i][1] == prev:
temp.append(y[i][0])
else:
prev = X[i][1]
temp2 = []
for elem in temp:
temp2.append((elem))
disp.append(temp2)
temp.clear()
temp.append(y[i][0])
disp.reverse()
plt.figure(figsize=(7, 7))
plt.xlabel("x")
plt.ylabel("y")
plt.title("PINN Prediction")
ax = plt.gca()
im = ax.imshow(disp, extent=(0, 1, 0, 1))
im.set_clim(0, 1)
ax.set_aspect(1)
divider = make_axes_locatable(ax)
width = ax.get_position().width
height = ax.get_position().height
cax = divider.append_axes("right", size="5%", pad=0.2)
plt.colorbar(im, cax=cax)
plt.show()
########################################
plt.figure()
X = x
y = PINNmodel.predict(x).tolist()
y_true = sol(x)
disp = []
prev = X[0][1]
temp = []
for i in range(len(y)):
if X[i][1] == prev:
temp.append(abs(y[i][0] - y_true[i][0]))
else:
prev = X[i][1]
temp2 = []
for elem in temp:
temp2.append((elem))
disp.append(temp2)
temp.clear()
temp.append(y[i][0])
disp.reverse()
plt.figure(figsize=(7, 7))
plt.xlabel("x")
plt.ylabel("y")
plt.title("PINN Absolute Error of u")
ax = plt.gca()
im = ax.imshow(disp, extent=(0, 1, 0, 1))
ax.set_aspect(1)
divider = make_axes_locatable(ax)
width = ax.get_position().width
height = ax.get_position().height
cax = divider.append_axes("right", size="5%", pad=0.2)
plt.colorbar(im, cax=cax)
plt.show()
# gPINN
a = 10
def f(x, y):
u_xx = (
16**a
* a
* (a * (1 - 2 * x) ** 2 - 2 * x**2 + 2 * x - 1)
* ((x - 1) * x * (y - 1) * y) ** a
/ ((x - 1) ** 2 * x**2)
)
u_yy = (
16**a
* a
* (a * (1 - 2 * y) ** 2 - 2 * y**2 + 2 * y - 1)
* ((x - 1) * x * (y - 1) * y) ** a
/ ((y - 1) ** 2 * y**2)
)
return -u_xx - u_yy
def f_x(x, y):
return -(
(
16**a
* a
* (2 * x - 1)
* ((x - 1) * x * (y - 1) * y) ** a
* (
-2 * a * (2 * a - 1) * (x - 1) ** 2 * x**2 * y
+ (a - 1) * a * (x - 1) ** 2 * x**2
+ (a - 1) * y**4 * (a * (1 - 2 * x) ** 2 - 2 * (x - 1) * x - 2)
- 2 * (a - 1) * y**3 * (a * (1 - 2 * x) ** 2 - 2 * (x - 1) * x - 2)
+ y**2
* (
(2 * a * (x - 1) * x + a) ** 2
+ a * (-2 * (x - 1) * x * ((x - 1) * x + 3) - 3)
+ 2 * (x - 1) * x
+ 2
)
)
)
/ ((x - 1) ** 3 * x**3 * (y - 1) ** 2 * y**2)
)
def f_y(x, y):
return -(
(
16**a
* a
* (2 * y - 1)
* ((x - 1) * x * (y - 1) * y) ** a
* (
(a - 1) * x**4 * (a * (1 - 2 * y) ** 2 - 2 * (y - 1) * y - 2)
- 2 * (a - 1) * x**3 * (a * (1 - 2 * y) ** 2 - 2 * (y - 1) * y - 2)
+ x**2
* (
(2 * a * (y - 1) * y + a) ** 2
+ a * (-2 * (y - 1) * y * ((y - 1) * y + 3) - 3)
+ 2 * (y - 1) * y
+ 2
)
- 2 * a * (2 * a - 1) * x * (y - 1) ** 2 * y**2
+ (a - 1) * a * (y - 1) ** 2 * y**2
)
)
/ ((x - 1) ** 2 * x**2 * (y - 1) ** 3 * y**3)
)
def output_transform(x, y):
x_in = x[:, 0:1]
y_in = x[:, 1:2]
return x_in * y_in * (1 - x_in) * (1 - y_in) * y
def pde(x, y):
x_in = x[:, 0:1]
y_in = x[:, 1:2]
du_xx = dde.grad.hessian(y, x, i=0, j=0)
du_yy = dde.grad.hessian(y, x, i=1, j=1)
du_xxx = dde.grad.jacobian(du_xx, x, j=0)
du_xxy = dde.grad.jacobian(du_xx, x, j=1)
du_yyy = dde.grad.jacobian(du_yy, x, j=1)
du_yyx = dde.grad.jacobian(du_yy, x, j=0)
return [
du_xx + du_yy + f(x_in, y_in),
du_xxx + du_yyx + f_x(x_in, y_in),
du_xxy + du_yyy + f_y(x_in, y_in),
]
geom = dde.geometry.Rectangle([0, 0], [1, 1])
data = dde.data.PDE(geom, pde, [], num_domain=400)
net = dde.maps.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
net.apply_output_transform(output_transform)
gPINNmodel = dde.Model(data, net)
gPINNmodel.compile("adam", lr=1.0e-3, loss_weights=[1, 1e-5, 1e-5])
losshistory, train_state = gPINNmodel.train(epochs=20000, callbacks=[])
def gen_test_x(num):
x = np.linspace(0, 1, num)
y = np.linspace(0, 1, num)
l = []
for i in range(len(y)):
for j in range(len(x)):
l.append([x[j], y[i]])
return np.array(l)
def sol(t):
x = t[:, 0:1]
y = t[:, 1:2]
return (16 * x * y * (1 - x) * (1 - y)) ** a
x = gen_test_x(100)
print(
"L2 relative error of u",
dde.metrics.l2_relative_error(sol(x), gPINNmodel.predict(x)),
)
##########################
plt.rcParams.update({"font.size": 22})
plt.figure()
X = gen_test_x(100)
y = gPINNmodel.predict(x).tolist()
disp = []
prev = X[0][1]
temp = []
for i in range(len(y)):
if X[i][1] == prev:
temp.append(y[i][0])
else:
prev = X[i][1]
temp2 = []
for elem in temp:
temp2.append((elem))
disp.append(temp2)
temp.clear()
temp.append(y[i][0])
disp.reverse()
plt.figure(figsize=(7, 7))
plt.xlabel("x")
plt.ylabel("y")
plt.title("gPINN Prediction")
ax = plt.gca()
im = ax.imshow(disp, extent=(0, 1, 0, 1))
im.set_clim(0, 1)
ax.set_aspect(1)
divider = make_axes_locatable(ax)
width = ax.get_position().width
height = ax.get_position().height
cax = divider.append_axes("right", size="5%", pad=0.2)
plt.colorbar(im, cax=cax)
plt.show()
########################################
plt.figure()
X = x
y = gPINNmodel.predict(x).tolist()
y_true = sol(x)
disp = []
prev = X[0][1]
temp = []
for i in range(len(y)):
if X[i][1] == prev:
temp.append(abs(y[i][0] - y_true[i][0]))
else:
prev = X[i][1]
temp2 = []
for elem in temp:
temp2.append((elem))
disp.append(temp2)
temp.clear()
temp.append(y[i][0])
disp.reverse()
plt.figure(figsize=(7, 7))
plt.xlabel("x")
plt.ylabel("y")
plt.title("gPINN Absolute Error of u")
ax = plt.gca()
im = ax.imshow(disp, extent=(0, 1, 0, 1))
im.set_clim(0, 0.004)
ax.set_aspect(1)
divider = make_axes_locatable(ax)
width = ax.get_position().width
height = ax.get_position().height
cax = divider.append_axes("right", size="5%", pad=0.2)
cbar = plt.colorbar(im, cax=cax, ticks=[0, 0.001, 0.002, 0.003, 0.004])
cbar.ax.set_yticklabels([0.000, 0.001, 0.002, 0.003, 0.004])
plt.show()