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_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 fpde(x, y, int_mat):
"""(D_{0+}^alpha + D_{1-}^alpha) u(x)"""
if isinstance(int_mat, (list, tuple)) and len(int_mat) == 3:
int_mat = tf.SparseTensor(*int_mat)
lhs = tf.sparse_tensor_dense_matmul(int_mat, y)
else:
lhs = tf.matmul(int_mat, y)
lhs /= 2 * tf.cos(alpha * np.pi / 2)
rhs = gamma(alpha0 + 2) * x
return lhs - rhs[:tf.size(lhs)]
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)