def main():
kf = tf.Variable(0.05)
D = tf.Variable(1.0)
def pde(x, y):
ca, cb = y[:, 0:1], y[:, 1:2]
dca_t = dde.grad.jacobian(y, x, i=0, j=1)
dca_xx = dde.grad.hessian(y, x, component=0, i=0, j=0)
dcb_t = dde.grad.jacobian(y, x, i=1, j=1)
dcb_xx = dde.grad.hessian(y, x, component=1, i=0, j=0)
eq_a = dca_t - 1e-3 * D * dca_xx + kf * ca * cb ** 2
eq_b = dcb_t - 1e-3 * D * dcb_xx + 2 * kf * ca * cb ** 2
return [eq_a, eq_b]
def fun_bc(x):
return 1 - x[:, 0:1]
def fun_init(x):
return np.exp(-20 * x[:, 0:1])
geom = dde.geometry.Interval(0, 1)
timedomain = dde.geometry.TimeDomain(0, 10)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
bc_a = dde.DirichletBC(
geomtime, fun_bc, lambda _, on_boundary: on_boundary, component=0
)
bc_b = dde.DirichletBC(
geomtime, fun_bc, lambda _, on_boundary: on_boundary, component=1
)
ic1 = dde.IC(geomtime, fun_init, lambda _, on_initial: on_initial, component=0)
ic2 = dde.IC(geomtime, fun_init, lambda _, on_initial: on_initial, component=1)
observe_x, Ca, Cb = gen_traindata()
ptset = dde.bc.PointSet(observe_x)
observe_y1 = dde.DirichletBC(
geomtime, ptset.values_to_func(Ca), lambda x, _: ptset.inside(x), component=0
)
observe_y2 = dde.DirichletBC(
geomtime, ptset.values_to_func(Cb), lambda x, _: ptset.inside(x), component=1
)
data = dde.data.TimePDE(
geomtime,
pde,
[bc_a, bc_b, ic1, ic2, observe_y1, observe_y2],
num_domain=2000,
num_boundary=100,
num_initial=100,
anchors=observe_x,
num_test=50000,
)
net = dde.maps.FNN([2] + [20] * 3 + [2], "tanh", "Glorot uniform")
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
variable = dde.callbacks.VariableValue(
[kf, D], period=1000, filename="variables.dat"
)
losshistory, train_state = model.train(epochs=80000, callbacks=[variable])
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
spatial_domain = dde.geometry.Rectangle(xmin=[-0.5, -0.5], xmax=[1, 1.5])
boundary_condition_u = dde.DirichletBC(spatial_domain,
u_func,
lambda _, on_boundary: on_boundary,
component=0)
boundary_condition_v = dde.DirichletBC(spatial_domain,
v_func,
lambda _, on_boundary: on_boundary,
component=1)
boundary_condition_right_p = dde.DirichletBC(spatial_domain,
p_func,
boundary_outflow,
component=2)
data = dde.data.TimePDE(
spatial_domain,
pde,
[
boundary_condition_u, boundary_condition_v,
boundary_condition_right_p
],
num_domain=2601,
num_boundary=400,
num_test=100000,
)
net = dde.maps.FNN([2] + 4 * [50] + [3], "tanh", "Glorot normal")
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
model.train(epochs=30000)
model.compile("L-BFGS-B")
losshistory, train_state = model.train()
X = spatial_domain.random_points(100000)
output = model.predict(X)
u_pred = output[:, 0]
v_pred = output[:, 1]
p_pred = output[:, 2]
u_exact = u_func(X).reshape(-1)
v_exact = v_func(X).reshape(-1)
p_exact = p_func(X).reshape(-1)
f = model.predict(X, operator=pde)
l2_difference_u = dde.metrics.l2_relative_error(u_exact, u_pred)
l2_difference_v = dde.metrics.l2_relative_error(v_exact, v_pred)
l2_difference_p = dde.metrics.l2_relative_error(p_exact, p_pred)
residual = np.mean(np.absolute(f))
print("Mean residual:", residual)
print("L2 relative error in u:", l2_difference_u)
print("L2 relative error in v:", l2_difference_v)
print("L2 relative error in p:", l2_difference_p)
def main():
C = tf.Variable(2.0)
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 - C * 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)
observe_x = np.vstack((np.linspace(-1, 1, num=10), np.full((10), 1))).T
ptset = dde.bc.PointSet(observe_x)
observe_y = dde.DirichletBC(geomtime,
ptset.values_to_func(func(observe_x)),
lambda x, _: ptset.inside(x))
data = dde.data.TimePDE(
geomtime,
1,
pde,
[bc, ic, observe_y],
num_domain=40,
num_boundary=20,
num_initial=10,
anchors=observe_x,
func=func,
num_test=10000,
)
layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
variable = dde.callbacks.VariableValue(C, period=1000)
losshistory, train_state = model.train(epochs=50000, callbacks=[variable])
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
def ode_system(x, y):
"""ODE system.
dy1/dx = y2
dy2/dx = -y1
"""
y1, y2, y3 = y[:, 0:1], y[:, 1:2], y[:, 2:]
dy1_x = tf.gradients(y1, x)[0]
dy2_x = tf.gradients(y2, x)[0]
dy3_x = tf.gradients(y3, x)[0]
return [dy1_x - y2, dy2_x + y1, dy3_x + y2]
def boundary(x, on_boundary):
return on_boundary and np.isclose(x[0], 0)
def func(x):
"""
y1 = sin(x)
y2 = cos(x)
"""
return np.hstack((np.sin(x), np.cos(x), -np.sin(x)))
geom = dde.geometry.Cuboid([0, 0, 0], [10, 10, 10])
bc1 = dde.DirichletBC(geom,
lambda x: np.sin(x[:, 0:1]),
boundary,
component=0)
bc2 = dde.DirichletBC(geom,
lambda x: np.cos(x[:, 0:1]),
boundary,
component=1)
bc3 = dde.DirichletBC(geom,
lambda x: np.sin(x[:, 0:1]),
boundary,
component=2)
data = dde.data.PDE(geom,
3,
ode_system, [bc1, bc2, bc3],
350,
64,
num_test=100)
layer_size = [3] + [50] * 3 + [3]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(epochs=20000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
def pde(x, y):
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:]
return -dy_xx - dy_yy - 1
def boundary(_, on_boundary):
return on_boundary
def func(x):
return np.zeros([len(x), 1])
geom = dde.geometry.Polygon([[0, 0], [1, 0], [1, -1], [-1, -1], [-1, 1], [0, 1]])
bc = dde.DirichletBC(geom, func, boundary)
data = dde.data.PDE(geom, pde, bc, num_domain=1200, num_boundary=120, num_test=1500)
net = dde.maps.FNN([2] + [50] * 4 + [1], "tanh", "Glorot uniform")
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
model.train(epochs=50000)
model.compile("L-BFGS-B")
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
def pde(x, y):
dy_x = tf.gradients(y, x)[0]
dy_xx = tf.gradients(dy_x, x)[0]
return dy_xx - 2
def boundary_l(x, on_boundary):
return on_boundary and np.isclose(x[0], -1)
def boundary_r(x, on_boundary):
return on_boundary and np.isclose(x[0], 1)
def func(x):
return (x + 1)**2
geom = dde.geometry.Interval(-1, 1)
bc_l = dde.DirichletBC(geom, func, boundary_l)
bc_r = dde.RobinBC(geom, lambda X, y: y, boundary_r)
data = dde.data.PDE(geom,
pde, [bc_l, bc_r],
16,
2,
solution=func,
num_test=100)
layer_size = [1] + [50] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
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 main():
geom = dde.geometry.Sphere([0, 0, 0], 1)
bc = dde.DirichletBC(geom, func, lambda _, on_boundary: on_boundary)
data = dde.data.FPDE(
geom,
fpde,
alpha,
bc,
[8, 8, 100],
num_domain=256,
num_boundary=1,
solution=func,
)
net = dde.maps.FNN([3] + [20] * 4 + [1], "tanh", "Glorot normal")
net.apply_output_transform(
lambda x, y: (1 - tf.reduce_sum(x**2, axis=1, keepdims=True)) * y)
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
losshistory, train_state = model.train(epochs=10000)
dde.saveplot(losshistory, train_state, issave=False, isplot=True)
X = geom.random_points(10000)
y_true = func(X)
y_pred = model.predict(X)
print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))
def main():
geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 0.99)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
bc = dde.DirichletBC(geomtime, lambda x: 0, lambda _, on_boundary: on_boundary)
ic = dde.IC(
geomtime, lambda x: -np.sin(np.pi * x[:, 0:1]), lambda _, on_initial: on_initial
)
data = dde.data.TimePDE(
geomtime, pde, [bc, ic], num_domain=2540, num_boundary=80, num_initial=160
)
net = dde.maps.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
model.train(epochs=15000)
model.compile("L-BFGS-B")
losshistory, train_state = model.train()
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)))
def main():
def pde(x, y):
dy_tt = dde.grad.hessian(y, x, i=1, j=1)
dy_xx = dde.grad.hessian(y, x, i=0, j=0)
return dy_tt - C**2 * dy_xx
def func(x):
x, t = np.split(x, 2, axis=1)
return np.sin(np.pi * x) * np.cos(C * np.pi * t) + np.sin(
A * np.pi * x) * np.cos(A * C * np.pi * t)
geom = dde.geometry.Interval(0, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
bc = dde.DirichletBC(geomtime, func, lambda _, on_boundary: on_boundary)
ic_1 = dde.IC(geomtime, func, lambda _, on_initial: on_initial)
# do not use dde.NeumannBC here, since `normal_derivative` does not work with temporal coordinate.
ic_2 = dde.OperatorBC(
geomtime,
lambda x, y, _: dde.grad.jacobian(y, x, i=0, j=1),
lambda x, _: np.isclose(x[1], 0),
)
data = dde.data.TimePDE(
geomtime,
pde,
[bc, ic_1, ic_2],
num_domain=360,
num_boundary=360,
num_initial=360,
solution=func,
num_test=10000,
)
layer_size = [2] + [100] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.STMsFFN(layer_size,
activation,
initializer,
sigmas_x=[1],
sigmas_t=[1, 10])
net.apply_feature_transform(lambda x: (x - 0.5) * 2 * np.sqrt(3))
model = dde.Model(data, net)
initial_losses = get_initial_loss(model)
loss_weights = 5 / initial_losses
model.compile(
"adam",
lr=0.001,
metrics=["l2 relative error"],
loss_weights=loss_weights,
decay=("inverse time", 2000, 0.9),
)
pde_residual_resampler = dde.callbacks.PDEResidualResampler(period=1)
losshistory, train_state = model.train(epochs=10000,
callbacks=[pde_residual_resampler],
display_every=500)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
def pde(x, y):
dy_xx = dde.grad.hessian(y, x, i=0, j=0)
dy_yy = dde.grad.hessian(y, x, i=1, j=1)
return -dy_xx - dy_yy - 1
def boundary(_, on_boundary):
return on_boundary
geom = dde.geometry.Polygon([[0, 0], [1, 0], [1, -1], [-1, -1], [-1, 1],
[0, 1]])
bc = dde.DirichletBC(geom, lambda x: 0, boundary)
data = dde.data.PDE(geom,
pde,
bc,
num_domain=1200,
num_boundary=120,
num_test=1500)
net = dde.maps.FNN([2] + [50] * 4 + [1], "tanh", "Glorot uniform")
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
model.train(epochs=50000)
model.compile("L-BFGS-B")
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
geom = dde.geometry.Interval(0, 1)
bc = dde.DirichletBC(geom, func, lambda _, on_boundary: on_boundary)
data = dde.data.PDE(
geom,
pde,
bc,
1280,
2,
train_distribution="pseudo",
solution=func,
num_test=10000,
)
layer_size = [1] + [100] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.MsFFN(layer_size, activation, initializer, sigmas=[1, 10])
model = dde.Model(data, net)
model.compile(
"adam",
lr=0.001,
metrics=["l2 relative error"],
decay=("inverse time", 2000, 0.9),
)
pde_residual_resampler = dde.callbacks.PDEResidualResampler(period=1)
model.train(epochs=20000, callbacks=[pde_residual_resampler])
dde.saveplot(model.losshistory, model.train_state, issave=True, isplot=True)
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 solve_poisson_with_dl(lightweight=False):
# geom = dde.geometry.Polygon([[0, 0], [1, 0], [1, -1], [-1, -1], [-1, 1], [0, 1]])
geom = dde.geometry.Rectangle(xmin=[0, 0], xmax=[1, 1])
# bc = dde.DirichletBC(geom, lambda x: 0, boundary)
bc = dde.DirichletBC(geom, _func, lambda _, on_boundary: on_boundary)
data = dde.data.PDE(
geom,
_pde,
bc,
num_domain=1200,
num_boundary=120,
num_test=10000,
# solution=_func,
)
# NN
layer_size = [2] + [128] + [256] + [512] + [256] + [128] + [1]
activation = 'relu'
initializer = 'Glorot uniform'
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
# Train NN
model.compile('adam', lr=0.0005)
losshistory, train_state = model.train(epochs=10000)
if not lightweight:
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
save_contour_from_model(model, 1, 1, 'poisson')
return model
def Poisson_PoliANN(Nd, Nb, Nh, Nl, sigma, opt, initializer):
# PDE Definition using the TensorFlow notation
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
# Definition of the boundary
def boundary(_, on_boundary):
return on_boundary
# Definition of the homogeneous Dirichlet
# boundary conditions
def func(x):
return np.zeros([len(x), 1])
# Geometry definition
geom = dde.geometry.Rectangle([0, 0], [1, 1])
# Imposition of the Dirichlet boundary condition
bc = dde.DirichletBC(geom, func, boundary)
# ANN model definition
data = dde.data.PDE(geom, pde, bc, num_domain=Nd, num_boundary=Nb)
net = dde.maps.FNN([2] + [Nl] * Nh + [1], sigma, initializer)
# Strict enforcement of the BCs for the unit square domain
# net.apply_output_transform(lambda x, y: \
# x[:,0:1] * (1 - x[:, 0:1]) * \
# x[:,1:2] * (1 - x[:, 1:2]) * y)
model = dde.Model(data, net)
# ANN model training
if opt == 'adam':
model.compile(opt, lr=0.001)
losshistory, train_state = model.train(epochs=10000)
elif opt == 'L-BFGS-B':
model.compile(opt)
losshistory, train_state = model.train()
elif opt == 'mixed':
model.compile('adam', lr=0.001)
model.train(epochs=5000)
model.compile('L-BFGS-B')
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
return model
def visualize_solution(nu):
# Build pseudo model
def pde(x, u):
u_x = dde.grad.jacobian(u, x, i=0, j=0)
u_t = dde.grad.jacobian(u, x, i=0, j=1)
u_xx = dde.grad.hessian(u, x, i=0, j=0)
return u_t + u * u_x - nu * u_xx
spatial_domain = dde.geometry.Interval(x_min, x_max)
temporal_domain = dde.geometry.TimeDomain(t_min, t_max)
spatio_temporal_domain = dde.geometry.GeometryXTime(
spatial_domain, temporal_domain)
boundary_condition = dde.DirichletBC(spatio_temporal_domain, lambda x: 0,
lambda _, on_boundary: on_boundary)
initial_condition = dde.IC(spatio_temporal_domain,
lambda x: -np.sin(np.pi * x[:, 0:1]),
lambda _, on_initial: on_initial)
data = dde.data.TimePDE(spatio_temporal_domain,
pde, [boundary_condition, initial_condition],
num_domain=domain_points,
num_boundary=boundary_points,
num_initial=initial_points,
num_test=test_points)
net = dde.maps.FNN([2] + hidden_layers * [hidden_units] + [1], "tanh",
"Glorot normal")
model = dde.Model(data, net)
# Load reference solution
file_name = 'Reference_Solutions/u_exact_nu_{}.mat'.format(nus[i])
data = scipy.io.loadmat(file_name)
u_exact = data['usol'].T
x_test, t_test = np.meshgrid(np.linspace(x_min, x_max, test_points_x),
np.linspace(t_min, t_max, test_points_t))
X = np.vstack((np.ravel(x_test), np.ravel(t_test))).T
# Reload model and make predictions
model_name = 'Neural_Networks/nu_{}/Burger_Equation_Source_Model_nu_{}-{}'.format(
nus[i], nus[i], epochs_source[i])
model.compile("adam", lr=learning_rate)
model.train(model_restore_path=model_name, epochs=0)
u_pred = model.predict(X).reshape(test_points_t, test_points_x)
f = model.predict(X, operator=pde)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x_test, t_test, u_pred)
ax.set_xlabel('location x')
ax.set_ylabel('time t')
ax.set_zlabel('u')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_nu_{}.png'.format(nus[i]),
dpi=300)
return
def main():
def pde(x, y):
dy_x = tf.gradients(y, x)[0]
dy_x, dy_t = dy_x[:, 0:1], dy_x[:, 1:2]
dy_xx = tf.gradients(dy_x, x)[0][:, 0:1]
return dy_t + y * dy_x - 0.01 / np.pi * dy_xx
geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 0.99)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
bc = dde.DirichletBC(geomtime, lambda x: np.zeros((len(x), 1)),
lambda _, on_boundary: on_boundary)
ic = dde.IC(geomtime, lambda x: -np.sin(np.pi * x[:, 0:1]),
lambda _, on_initial: on_initial)
data = dde.data.TimePDE(geomtime,
1,
pde, [bc, ic],
num_domain=2500,
num_boundary=100,
num_initial=160)
net = dde.maps.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
model = dde.Model(data, net)
model.compile("adam", lr=1.0e-3)
model.train(epochs=10000)
model.compile("L-BFGS-B")
model.train()
X = geomtime.random_points(100000)
err = 1
while err > 0.005:
f = model.predict(X, operator=pde)
err_eq = np.absolute(f)
err = np.mean(err_eq)
print("Mean residual: %.3e" % (err))
x_id = np.argmax(err_eq)
print("Adding new point:", X[x_id], "\n")
data.add_anchors(X[x_id])
early_stopping = dde.callbacks.EarlyStopping(min_delta=1e-4,
patience=2000)
model.compile("adam", lr=1e-3)
model.train(epochs=10000,
disregard_previous_best=True,
callbacks=[early_stopping])
model.compile("L-BFGS-B")
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
X, y_true = gen_testdata()
y_pred = model.predict(X)
print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))
def main():
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 func(x):
return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 4:])
def func2(x):
return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 4:]),0
# geom = dde.geometry.Interval(-1, 1)
geom = dde.geometry.Rectangle
geom = dde.geometry.Hypercube([-1]*4,[1]*4)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)
bc = dde.DirichletBC(geomtime, func, lambda _, on_boundary: on_boundary,component=0)
ic = dde.IC(geomtime, func, lambda _, on_initial: on_initial,component=0)
ic2 = dde.IC(geomtime,lambda shit: 1, lambda _, on_initial: on_initial,component=1)
# observe_x = np.vstack((np.linspace(-1, 1, num=10), np.full((10), 1))).T
# ptset = dde.bc.PointSet(observe_x)
# observe_y = dde.DirichletBC(
# geomtime, ptset.values_to_func(func(observe_x)), lambda x, _: ptset.inside(x)
# )
data = dde.data.TimePDE(
geomtime,
pde,
[bc, ic,ic2],
num_domain=4000,
num_boundary=1,
num_initial=100,
)
layer_size = [5] + [32] * 3 + [2]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
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 train_source_model(nu):
path = Path('Neural_Networks', 'nu_{}'.format(nus[i]))
if path.exists() and path.is_dir():
shutil.rmtree(path)
os.mkdir(path)
def pde(x, u):
du_xx = dde.grad.hessian(u, x)
return du_xx + nu*np.pi ** 2 * tf.sin(np.pi * x)
def func(x):
return nu*np.sin(np.pi * x)
spatial_domain = dde.geometry.Interval(x_min, x_max)
boundary_condition = dde.DirichletBC(spatial_domain, lambda x: 0, lambda _, on_boundary: on_boundary)
data = dde.data.PDE(spatial_domain, pde, [boundary_condition],
num_domain=domain_points, num_boundary=boundary_points, solution = func, num_test=test_points)
net = dde.maps.FNN([1] + hidden_layers * [hidden_units] + [1], "tanh", "Glorot normal")
model = dde.Model(data, net)
model_name = 'Neural_Networks/nu_{}/Poisson_Equation_Source_Model_nu_{}'.format(nus[i], nus[i])
start = time.time()
model.compile("adam", lr=1e-3)
history, train_state = model.train(epochs=number_of_epochs, model_save_path=model_name)
end = time.time()
length = end - start
X = np.linspace(x_min, x_max, test_points).reshape(-1, 1)
u_pred = model.predict(X)
u_exact = func(X)
f = model.predict(X, operator=pde)
figure_name = 'Predictions/Predicted_solution_nu_{}'.format(nu)
plt.plot(X, u_exact, color = 'blue', label = 'exact solution')
plt.plot(X, u_pred, color = 'red', linestyle ='--', label = 'predicted solution')
plt.xlabel(r'location $x$')
plt.ylabel(r'$u$')
plt.legend(loc="upper left")
plt.tight_layout()
plt.savefig(figure_name, dpi = 600)
residual = np.mean(np.absolute(f))
l2_difference = dde.metrics.l2_relative_error(u_exact, u_pred)
def main():
alpha0 = 1.8
alpha = tf.Variable(1.5)
def fpde(x, y, int_mat):
"""\int_theta D_theta^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 = lhs[:, 0]
lhs *= -tf.exp(
tf.lgamma((1 - alpha) / 2) + tf.lgamma(
(2 + alpha) / 2)) / (2 * np.pi**1.5)
x = x[:tf.size(lhs)]
rhs = (2**alpha0 * gamma(2 + alpha0 / 2) * gamma(1 + alpha0 / 2) *
(1 - (1 + alpha0 / 2) * tf.reduce_sum(x**2, axis=1)))
return lhs - rhs
def func(x):
return (1 - np.linalg.norm(x, axis=1, keepdims=True)**2)**(1 +
alpha0 / 2)
geom = dde.geometry.Disk([0, 0], 1)
observe_x = geom.random_points(30)
ptset = dde.bc.PointSet(observe_x)
observe_y = dde.DirichletBC(geom, ptset.values_to_func(func(observe_x)),
lambda x, _: ptset.inside(x))
data = dde.data.FPDE(
geom,
fpde,
alpha,
observe_y,
[8, 100],
num_domain=64,
anchors=observe_x,
solution=func,
)
net = dde.maps.FNN([2] + [20] * 4 + [1], "tanh", "Glorot normal")
net.apply_output_transform(
lambda x, y: (1 - tf.reduce_sum(x**2, axis=1, keepdims=True)) * y)
model = dde.Model(data, net)
model.compile("adam", lr=1e-3, loss_weights=[1, 100])
variable = dde.callbacks.VariableValue(alpha, period=1000)
losshistory, train_state = model.train(epochs=10000, callbacks=[variable])
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
alpha = 1.5
def fpde(x, y, int_mat):
"""(D_{0+}^alpha + D_{1-}^alpha) u(x) = f(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)
rhs = (gamma(4) / gamma(4 - alpha) * (x**(3 - alpha) +
(1 - x)**(3 - alpha)) -
3 * gamma(5) / gamma(5 - alpha) * (x**(4 - alpha) +
(1 - x)**(4 - alpha)) +
3 * gamma(6) / gamma(6 - alpha) * (x**(5 - alpha) +
(1 - x)**(5 - alpha)) -
gamma(7) / gamma(7 - alpha) * (x**(6 - alpha) +
(1 - x)**(6 - alpha)))
# lhs /= 2 * np.cos(alpha * np.pi / 2)
# rhs = gamma(alpha + 2) * x
return lhs - rhs[:tf.size(lhs)]
def func(x):
# return x * (np.abs(1 - x**2)) ** (alpha / 2)
return x**3 * (1 - x)**3
geom = dde.geometry.Interval(0, 1)
bc = dde.DirichletBC(geom, func, lambda _, on_boundary: on_boundary)
# Static auxiliary points
data = dde.data.FPDE(geom,
fpde,
alpha,
bc, [101],
meshtype="static",
solution=func)
# Dynamic auxiliary points
# data = dde.data.FPDE(
# geom, fpde, alpha, bc, [100], meshtype="dynamic", num_domain=20, num_boundary=2, solution=func, num_test=100
# )
net = dde.maps.FNN([1] + [20] * 4 + [1], "tanh", "Glorot normal")
net.apply_output_transform(lambda x, y: x * (1 - x) * y)
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
losshistory, train_state = model.train(epochs=10000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
"""
main function for differential function with q
"""
dde.config.real.set_float64()
# geometry part
tmax = 1
Qmax = 10 / C.sigma
Xmin = []
Xmax = []
for i in range(1, C.N + 1):
Xmin.append(-Qmax)
Xmax.append(Qmax)
geom = dde.geometry.Hypercube(Xmin, Xmax)
# geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, tmax)
geom = dde.geometry.GeometryXTime(geom, timedomain)
x_initial = np.random.rand(13, C.N + 1)
xtest = tf.convert_to_tensor(x_initial)
ytest = np.random.rand(13, C.sizeRho)
ytest = tf.convert_to_tensor(ytest)
# Initial conditions
ic = []
for j in range(0, (C.N + 1)**2):
ic.append(
dde.IC(geom, lambda X: initialState(X, j), boundary, component=j))
# test
# print(initialState(x_initial,j))
# print(SLE_q(xtest,ytest))
bc = dde.DirichletBC(geom, lambda _: 0, lambda _, on_boundary: on_boundary)
ic.append(bc)
# data
data = dde.data.TimePDE(geom,
lambda x, y: SLE_q(x, y, C),
ic,
num_domain=4000,
num_boundary=0,
num_initial=100,
num_test=None)
layer_size = [C.N + 1] + [50] * 3 + [(C.N + 1)**2]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
model.compile("L-BFGS-B")
losshistory, train_state = model.train(epochs=600000,
callbacks=ModelCheckpoint)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
def ddy(x, y):
dy_x = tf.gradients(y, x)[0]
return tf.gradients(dy_x, x)[0]
def dddy(x, y):
return tf.gradients(ddy(x, y), x)[0]
def pde(x, y):
dy_xxxx = tf.gradients(dddy(x, y), x)[0]
return dy_xxxx + 1
def boundary_l(x, on_boundary):
return on_boundary and np.isclose(x[0], 0)
def boundary_r(x, on_boundary):
return on_boundary and np.isclose(x[0], 1)
def func(x):
return -x ** 4 / 24 + x ** 3 / 6 - x ** 2 / 4
geom = dde.geometry.Interval(0, 1)
zero_func = lambda x: np.zeros((len(x), 1))
bc1 = dde.DirichletBC(geom, zero_func, boundary_l)
bc2 = dde.NeumannBC(geom, zero_func, boundary_l)
bc3 = dde.OperatorBC(geom, lambda x, y, _: ddy(x, y), boundary_r)
bc4 = dde.OperatorBC(geom, lambda x, y, _: dddy(x, y), boundary_r)
data = dde.data.PDE(
geom,
1,
pde,
[bc1, bc2, bc3, bc4],
num_domain=10,
num_boundary=2,
func=func,
num_test=100,
)
layer_size = [1] + [20] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
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 main():
geom = dde.geometry.Interval(0, 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)
# Static auxiliary points
data = dde.data.TimeFPDE(
geomtime,
fpde,
alpha,
[bc, ic],
[52],
meshtype="static",
num_domain=400,
solution=func,
)
# Dynamic auxiliary points
# data = dde.data.TimeFPDE(
# geomtime,
# fpde,
# alpha,
# [bc, ic],
# [100],
# num_domain=20,
# num_boundary=1,
# num_initial=1,
# solution=func,
# num_test=50,
# )
net = dde.maps.FNN([2] + [20] * 4 + [1], "tanh", "Glorot normal")
net.apply_output_transform(
lambda x, y: x[:, 0:1] * (1 - x[:, 0:1]) * x[:, 1:2] * y
+ x[:, 0:1] ** 3 * (1 - x[:, 0:1]) ** 3
)
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
losshistory, train_state = model.train(epochs=10000)
dde.saveplot(losshistory, train_state, issave=False, isplot=True)
X = geomtime.random_points(1000)
y_true = func(X)
y_pred = model.predict(X)
print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))
def main():
def ide(x, y, int_mat):
rhs = tf.matmul(int_mat, y)
lhs1 = tf.gradients(y, x)[0]
return (lhs1 + y)[:tf.size(rhs)] - rhs
def kernel(x, s):
return np.exp(s - x)
def boundary(x, on_boundary):
return on_boundary and np.isclose(x[0], 0)
def func(x):
return np.exp(-x) * np.cosh(x)
geom = dde.geometry.Interval(0, 5)
bc = dde.DirichletBC(geom, func, boundary)
quad_deg = 20
data = dde.data.IDE(
geom,
ide,
bc,
quad_deg,
kernel=kernel,
num_domain=10,
num_boundary=2,
train_distribution="uniform",
)
layer_size = [1] + [20] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("L-BFGS-B")
model.train()
X = geom.uniform_points(100)
y_true = func(X)
y_pred = model.predict(X)
print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
plt.figure()
plt.plot(X, y_true, "-")
plt.plot(X, y_pred, "o")
plt.show()
np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))
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)
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=20,
num_initial=10,
train_distribution="pseudo",
solution=func,
num_test=10000,
)
layer_size = [2] + [32] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
for _ in range(5):
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
model.train(epochs=2000)
print("epoch = {}, resample train points...".format(model.train_state.epoch))
data.resample_train_points()
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
losshistory, train_state = model.train(epochs=2000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
alpha = 1.8
def fpde(x, y, int_mat):
"""\int_theta D_theta^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 = lhs[:, 0]
lhs *= gamma((1 - alpha) / 2) * gamma(
(2 + alpha) / 2) / (2 * np.pi**1.5)
x = x[:tf.size(lhs)]
rhs = (2**alpha * gamma(2 + alpha / 2) * gamma(1 + alpha / 2) *
(1 - (1 + alpha / 2) * tf.reduce_sum(x**2, axis=1)))
return lhs - rhs
def func(x):
return (np.abs(1 - np.linalg.norm(x, axis=1, keepdims=True)**2))**(
1 + alpha / 2)
geom = dde.geometry.Disk([0, 0], 1)
bc = dde.DirichletBC(geom, func, lambda _, on_boundary: on_boundary)
data = dde.data.FPDE(geom,
fpde,
alpha,
bc, [8, 100],
num_domain=100,
num_boundary=1,
solution=func)
net = dde.maps.FNN([2] + [20] * 4 + [1], "tanh", "Glorot normal")
net.apply_output_transform(
lambda x, y: (1 - tf.reduce_sum(x**2, axis=1, keepdims=True)) * y)
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
losshistory, train_state = model.train(epochs=20000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
X = geom.random_points(1000)
y_true = func(X)
y_pred = model.predict(X)
print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))
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 main():
def pde(x, y):
dy_xx = dde.grad.hessian(y, x)
return -dy_xx - np.pi ** 2 * tf.sin(np.pi * x)
def boundary(x, on_boundary):
return on_boundary
def func(x):
return np.sin(np.pi * x)
geom = dde.geometry.Interval(-1, 1)
bc = dde.DirichletBC(geom, func, boundary)
data = dde.data.PDE(geom, pde, bc, 16, 2, solution=func, num_test=100)
layer_size = [1] + [50] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
checkpointer = dde.callbacks.ModelCheckpoint(
"model/model.ckpt", verbose=1, save_better_only=True
)
# ImageMagick (https://imagemagick.org/) is required to generate the movie.
movie = dde.callbacks.MovieDumper(
"model/movie", [-1], [1], period=100, save_spectrum=True, y_reference=func
)
losshistory, train_state = model.train(
epochs=10000, callbacks=[checkpointer, movie]
)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
# Plot PDE residual
model.restore("model/model.ckpt-" + str(train_state.best_step), verbose=1)
x = geom.uniform_points(1000, True)
y = model.predict(x, operator=pde)
plt.figure()
plt.plot(x, y)
plt.xlabel("x")
plt.ylabel("PDE residual")
plt.show()
def main():
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 boundary(x, on_boundary):
return on_boundary and np.isclose(x[0], 0)
def func(x):
"""
x: array_like, N x D_in
y: array_like, N x D_out
"""
return np.sin(2 * np.pi * x)
geom = dde.geometry.Interval(0, 1)
bc = dde.DirichletBC(geom, func, boundary)
quad_deg = 16
data = dde.data.IDE(geom, ide, bc, quad_deg, num_domain=16, num_boundary=2)
layer_size = [1] + [20] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
model.train(epochs=10000)
X = geom.uniform_points(100, True)
y_true = func(X)
y_pred = model.predict(X)
print("L2 relative error:", dde.metrics.l2_relative_error(y_true, y_pred))
plt.figure()
plt.plot(X, y_true, "-")
plt.plot(X, y_pred, "o")
plt.show()
np.savetxt("test.dat", np.hstack((X, y_true, y_pred)))
def main():
geom = dde.geometry.Polygon([[0, 0], [1, 0], [1, -1], [-1, -1], [-1, 1],
[0, 1]])
bc = dde.DirichletBC(geom, lambda x: 0, boundary)
data = dde.data.PDE(geom,
pde,
bc,
num_domain=1200,
num_boundary=120,
num_test=1500)
net = dde.maps.FNN([2] + [50] * 4 + [1], "tanh", "Glorot uniform")
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
model.train(epochs=50000)
model.compile("L-BFGS-B")
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave=True, isplot=True)