def feature_transform(t):
t = 0.01 * t
return tf.concat(
(t, tf.sin(t), tf.sin(2 * t), tf.sin(3 * t), tf.sin(
4 * t), tf.sin(5 * t)),
axis=1,
)
def pde(x, y):
dy_xx = dde.grad.hessian(y, x)
return (
dy_xx
+ (np.pi * A) ** 2 * tf.sin(np.pi * A * x)
+ 0.1 * (np.pi * B) ** 2 * tf.sin(np.pi * B * x)
)
def pde(x, y):
dy_x = tf.gradients(y, x)[0]
dy_x, dy_t = dy_x[:, 0:1], dy_x[:, 1:]
dy_xx = tf.gradients(dy_x, x)[0][:, 0:1]
return (
dy_t - dy_xx + tf.exp(-x[:, 1:]) *
(tf.sin(np.pi * x[:, 0:1]) - np.pi**2 * tf.sin(np.pi * x[:, 0:1])))
def pde(x, y):
J = dde.grad.Jacobian(y, x)
dy_t = J(j=1)
H = dde.grad.Hessian(y, x, grad_y=J())
dy_xx = H(j=0)
return (
dy_t - dy_xx + tf.exp(-x[:, 1:]) *
(tf.sin(np.pi * x[:, 0:1]) - np.pi**2 * tf.sin(np.pi * x[:, 0:1])))
def pde(x, y):
dy_t = dde.grad.jacobian(y, x, i=0, j=1)
dy_xx = dde.grad.hessian(y, x, i=0, j=0)
return (
dy_t
- dy_xx
+ tf.exp(-x[:, 1:])
* (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
)
def pde(x, y):
dy_t = dde.grad.jacobian(y, x, i=0, j=1)
dy_xx = dde.grad.hessian(y, x, i=0, j=0)
# Backend tensorflow.compat.v1 or tensorflow
return (
dy_t
- dy_xx
+ tf.exp(-x[:, 1:])
* (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
)
def pde(x, y):
dy_t = dde.grad.jacobian(y, x,i=0, j=4)
dy_xx = dde.grad.hessian(y, x,component=0 , j=0)
# dy_xx = dde.grad.hessian(y, x , j=0)
return (
dy_t
- dy_xx
+ tf.exp(-x[:, 4:])
* (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1])),
x[:, 0:1] * 0,
)
def pde(x, y):
f = 2 * (np.pi**2) * tf.sin(np.pi * x[:, 0:1]) \
* tf.sin(np.pi * x[:, 1:2])
# Definition of the spatial derivatives
dy_x = tf.gradients(y, x)[0]
dy_x, dy_y = dy_x[:, 0:1], dy_x[:, 1:]
dy_xx = tf.gradients(dy_x, x)[0][:, 0:1]
dy_yy = tf.gradients(dy_y, x)[0][:, 1:]
# Definition of the Poisson equation
return dy_xx + dy_yy + f
def main():
def pde(x, y):
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
def solution(x):
r, theta = x[:, 0:1], x[:, 1:]
return r * np.cos(theta)
geom = dde.geometry.Rectangle(xmin=[0, 0], xmax=[1, 2 * np.pi])
bc_rad = dde.DirichletBC(
geom,
lambda x: np.cos(x[:, 1:2]),
lambda x, on_boundary: on_boundary and np.isclose(x[0], 1),
)
data = dde.data.PDE(geom,
pde,
bc_rad,
num_domain=2540,
num_boundary=80,
solution=solution)
net = dde.maps.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
# Use [r*sin(theta), r*cos(theta)] as features,
# so that the network is automatically periodic along the theta coordinate.
net.apply_feature_transform(lambda x: tf.concat(
[x[:, 0:1] * tf.sin(x[:, 1:2]), x[:, 0:1] * tf.cos(x[:, 1:2])], axis=1)
)
model = dde.Model(data, net)
model.compile("adam", lr=1e-3, metrics=["l2 relative error"])
losshistory, train_state = model.train(epochs=15000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def ide(x, y, int_mat):
"""int_0^x y(t)dt
"""
lhs1 = tf.matmul(int_mat, y)
lhs2 = tf.gradients(y, x)[0]
rhs = 2 * np.pi * tf.cos(2 * np.pi * x) + tf.sin(np.pi * x) ** 2 / np.pi
return lhs1 + (lhs2 - rhs)[: tf.size(lhs1)]
def main():
def pde(x, y):
dy_x = tf.gradients(y, x)[0]
dy_x, dy_t = dy_x[:, 0:1], dy_x[:, 1:]
dy_xx = tf.gradients(dy_x, x)[0][:, 0:1]
return (
dy_t
- dy_xx
+ tf.exp(-x[:, 1:])
* (tf.sin(np.pi * x[:, 0:1]) - np.pi ** 2 * tf.sin(np.pi * x[:, 0:1]))
)
def func(x):
return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])
geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
bc = dde.DirichletBC(geomtime, func, lambda _, on_boundary: on_boundary)
ic = dde.IC(geomtime, func, lambda _, on_initial: on_initial)
data = dde.data.TimePDE(
geomtime,
pde,
[bc, ic],
num_domain=40,
num_boundary=1,
num_initial=1,
solution=func,
num_test=10000,
)
layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
net.outputs_modify(
lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + tf.sin(np.pi * x[:, 0:1])
)
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(epochs=10000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def feature_transform(t):
return tf.concat(
(
t,
tf.sin(t),
tf.sin(2 * t),
tf.sin(3 * t),
tf.sin(4 * t),
tf.sin(5 * t),
tf.sin(6 * t),
),
axis=1,
)
def main():
def pde(x, y):
dy_t = dde.grad.jacobian(y, x, i=0, j=1)
dy_xx = dde.grad.hessian(y, x, i=0, j=0)
return (
dy_t - dy_xx + tf.exp(-x[:, 1:]) *
(tf.sin(np.pi * x[:, 0:1]) - np.pi**2 * tf.sin(np.pi * x[:, 0:1])))
def func(x):
return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])
geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
data = dde.data.TimePDE(
geomtime,
pde,
[],
num_domain=40,
solution=func,
num_test=10000,
)
layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
net.apply_output_transform(lambda x, y: x[:, 1:2] * (1 - x[:, 0:1]**2) * y
+ tf.sin(np.pi * x[:, 0:1]))
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(epochs=10000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def pde(x, y):
dy_xx = dde.grad.hessian(y, x)
return -dy_xx - np.pi ** 2 * tf.sin(np.pi * x)
return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])
geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
data = dde.data.TimePDE(geomtime,
pde, [],
num_domain=40,
solution=func,
num_test=10000)
layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
net.apply_output_transform(
# Backend tensorflow.compat.v1 or tensorflow
lambda x, y: x[:, 1:2] * (1 - x[:, 0:1]**2) * y + tf.sin(np.pi * x[:, 0:1])
# Backend pytorch
# lambda x, y: x[:, 1:2] * (1 - x[:, 0:1] ** 2) * y + torch.sin(np.pi * x[:, 0:1])
)
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(epochs=10000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def pde(x, y):
dy_x = tf.gradients(y, x)[0]
dy_xx = tf.gradients(dy_x, x)[0]
return -dy_xx - np.pi**2 * tf.sin(np.pi * x)
def pde(x, y):
dy_xx = dde.grad.hessian(y, x)
# Use tf.sin for backend tensorflow.compat.v1 or tensorflow
return -dy_xx - np.pi**2 * tf.sin(np.pi * x)
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)
