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poisson_1d.py
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poisson_1d.py
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from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
import deepxde as dde
from deepxde.backend import tf
# PINN
def PINNpde(x, y):
dy_xx = dde.grad.hessian(y, x)
f = 8 * tf.sin(8 * x)
for i in range(1, 5):
f += i * tf.sin(i * x)
return -dy_xx - f
def func(x):
sol = x + 1 / 8 * np.sin(8 * x)
for i in range(1, 5):
sol += 1 / i * np.sin(i * x)
return sol
geom = dde.geometry.Interval(0, np.pi)
data = dde.data.PDE(geom, PINNpde, [], 15, 0, "uniform", solution=func, num_test=100)
layer_size = [1] + [20] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
def output_transform(x, y):
return x + tf.math.tanh(x) * tf.math.tanh(np.pi - x) * y
net.apply_output_transform(output_transform)
PINNmodel = dde.Model(data, net)
PINNmodel.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = PINNmodel.train(epochs=20000)
dde.saveplot(losshistory, train_state, issave=True, isplot=False)
# gPINN
def gPINNpde(x, y):
dy_xx = dde.grad.hessian(y, x)
dy_xxx = dde.grad.jacobian(dy_xx, x)
f = 8 * tf.sin(8 * x)
for i in range(1, 5):
f += i * tf.sin(i * x)
df_x = (
tf.cos(x)
+ 4 * tf.cos(2 * x)
+ 9 * tf.cos(3 * x)
+ 16 * tf.cos(4 * x)
+ 64 * tf.cos(8 * x)
)
return [-dy_xx - f, -dy_xxx - df_x]
geom = dde.geometry.Interval(0, np.pi)
data = dde.data.PDE(geom, gPINNpde, [], 15, 0, "uniform", solution=func, num_test=100)
layer_size = [1] + [20] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
def output_transform(x, y):
return x + tf.math.tanh(x) * tf.math.tanh(np.pi - x) * y
net.apply_output_transform(output_transform)
gPINNmodel = dde.Model(data, net)
gPINNmodel.compile(
"adam", lr=0.001, metrics=["l2 relative error"], loss_weights=[1, 0.01]
)
losshistory, train_state = gPINNmodel.train(epochs=20000)
dde.saveplot(losshistory, train_state, issave=True, isplot=False)
# plots
x = geom.uniform_points(1000)
plt.figure()
plt.plot(x, func(x), label="Exact", color="black")
plt.plot(x, PINNmodel.predict(x), label="PINN", color="blue", linestyle="dashed")
plt.plot(
x, gPINNmodel.predict(x), label="gPINN, w = 0.01", color="red", linestyle="dashed"
)
plt.legend(frameon=False)
x = geom.uniform_points(15, boundary=False)
plt.plot(x, func(x), color="black", marker="o", linestyle="none")
plt.xlabel("x")
plt.ylabel("u")
x = geom.uniform_points(1000)
def du_x(x):
return 1 + np.cos(x) + np.cos(2 * x) + np.cos(3 * x) + np.cos(4 * x) + np.cos(8 * x)
plt.figure()
plt.plot(x, du_x(x), label="Exact", color="black")
plt.plot(
x,
PINNmodel.predict(x, operator=lambda x, y: dde.grad.jacobian(y, x)),
label="PINN",
color="blue",
linestyle="dashed",
)
plt.plot(
x,
gPINNmodel.predict(x, operator=lambda x, y: dde.grad.jacobian(y, x)),
label="gPINN, w = 0.01",
color="red",
linestyle="dashed",
)
x = geom.uniform_points(15, boundary=False)
plt.plot(x, du_x(x), color="black", marker="o", linestyle="none")
plt.legend(frameon=False)
plt.xlabel("x")
plt.ylabel("u'")
plt.show()