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():
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 main():
C1 = tf.Variable(1.0)
C2 = tf.Variable(1.0)
C3 = tf.Variable(1.0)
def Lorenz_system(x, y):
"""Lorenz system.
dy1/dx = 10 * (y2 - y1)
dy2/dx = y1 * (28 - y3) - y2
dy3/dx = y1 * y2 - 8/3 * y3
"""
y1, y2, y3 = y[:, 0:1], y[:, 1:2], y[:, 2:]
dy1_x = dde.grad.jacobian(y, x, i=0)
dy2_x = dde.grad.jacobian(y, x, i=1)
dy3_x = dde.grad.jacobian(y, x, i=2)
return [
dy1_x - C1 * (y2 - y1),
dy2_x - y1 * (C2 - y3) + y2,
dy3_x - y1 * y2 + C3 * y3,
]
def boundary(_, on_initial):
return on_initial
geom = dde.geometry.TimeDomain(0, 3)
# Initial conditions
ic1 = dde.IC(geom, lambda X: -8, boundary, component=0)
ic2 = dde.IC(geom, lambda X: 7, boundary, component=1)
ic3 = dde.IC(geom, lambda X: 27, boundary, component=2)
# Get the train data
observe_t, ob_y = gen_traindata()
observe_y0 = dde.PointSetBC(observe_t, ob_y[:, 0:1], component=0)
observe_y1 = dde.PointSetBC(observe_t, ob_y[:, 1:2], component=1)
observe_y2 = dde.PointSetBC(observe_t, ob_y[:, 2:3], component=2)
data = dde.data.PDE(
geom,
Lorenz_system,
[ic1, ic2, ic3, observe_y0, observe_y1, observe_y2],
num_domain=400,
num_boundary=2,
anchors=observe_t,
)
net = dde.maps.FNN([1] + [40] * 3 + [3], "tanh", "Glorot uniform")
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
variable = dde.callbacks.VariableValue([C1, C2, C3],
period=600,
filename="variables.dat")
losshistory, train_state = model.train(epochs=60000, callbacks=[variable])
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
#sir params
beta = 1 #infection
gamma = .2 #removal
N = 150 #population
initial_infected = 20
#x is time, y is odes
def odes(x, y):
s, i, r = y[:, 0:1], y[:, 1:2], y[:, 2:3]
ds_x = tf.gradients(s, x)[0]
di_x = tf.gradients(i, x)[0]
dr_x = tf.gradients(r, x)[0]
de1 = ds_x + beta * i * s / N
de2 = di_x - (beta * i * s / N) + (gamma * i)
de3 = dr_x - gamma * i
# de1 = tf.gradients(ds_x, x)[0] - s
# de2 = tf.gradients(di_x, x)[0] - i
# de3 = tf.gradients(dr_x, x)[0] - r
return [de1, de2, de3]
def boundary(_, on_initial):
return on_initial
def func(x):
return np.hstack((0 * np.cos(x), 0 * np.cos(x), 0 * np.cos(x)))
#return np.hstack(((N-initial_infected)*np.cos(x), initial_infected*np.cos(x), 0*np.cos(x)))
geom = dde.geometry.TimeDomain(0, 15)
ic1 = dde.IC(geom,
lambda x: (N - initial_infected) * np.ones(x.shape),
boundary,
component=0)
ic2 = dde.IC(geom,
lambda x: initial_infected * np.ones(x.shape),
boundary,
component=1)
ic3 = dde.IC(geom, lambda x: 0 * np.ones(x.shape), boundary, component=2)
data = dde.data.PDE(geom, odes, [ic1, ic2, ic3], 1000, 3)
layer_size = [1] + [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)
losshistory, train_state = model.train(epochs=20000)
#dde.saveplot(losshistory, train_state, issave=True, isplot=True)
plot_best_state(train_state)
plot_loss_history(losshistory)
plt.show()
def main():
geom = dde.geometry.TimeDomain(0, 5)
ic = dde.IC(geom, func, lambda _, on_initial: on_initial)
quad_deg = 20
data = dde.data.IDE(
geom,
ide,
ic,
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_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():
"""
main program
haotian song 2020/12/23
"""
geom = dde.geometry.TimeDomain(0, 3)
y0 = InitialCondition()
a = dde.config.real.set_complex128()
ic = []
for j in range(0, len(y0)):
ic.append(dde.IC(geom, lambda X: y0, boundary, component=j))
# BoundaryCondition = dde.OperatorBC(geom,boundaryFunction,on_boundary=on_boundary)
data = dde.data.PDE(geom,
SLE_DL,
ic,
num_domain=400,
num_boundary=2,
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)
variable = dde.callbacks.VariableValue([C1, C2, C3],
period=600,
filename="variables.dat")
losshistory, train_state = model.train(epochs=60000, callbacks=[variable])
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
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 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 ode_system(x, y):
"""ODE system.
dy1/dx = y2
dy2/dx = -y1
"""
y1, y2 = y[:, 0:1], y[:, 1:]
dy1_x = tf.gradients(y1, x)[0]
dy2_x = tf.gradients(y2, x)[0]
return [dy1_x - y2, dy2_x + y1]
def boundary(_, on_initial):
return on_initial
def func(x):
"""
y1 = sin(x)
y2 = cos(x)
"""
return np.hstack((np.sin(x), np.cos(x)))
geom = dde.geometry.TimeDomain(0, 10)
ic1 = dde.IC(geom, np.sin, boundary, component=0)
ic2 = dde.IC(geom, np.cos, boundary, component=1)
data = dde.data.PDE(geom,
ode_system, [ic1, ic2],
35,
2,
solution=func,
num_test=100)
layer_size = [1] + [50] * 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=20000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
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():
"""
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():
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 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():
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 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 func(x):
return np.exp(-x) * np.cosh(x)
geom = dde.geometry.TimeDomain(0, 5)
ic = dde.IC(geom, func, lambda _, on_initial: on_initial)
quad_deg = 20
data = dde.data.IDE(
geom,
ide,
ic,
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, j=1)
dy_xx = dde.grad.hessian(y, x, 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:])
def func2(x):
return 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=4000,
num_boundary=20,
num_initial=10,
solution=func,
num_test=None,
)
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"])
losshistory, train_state = model.train(epochs=10000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
model.save('./diffusion_1d_model', verbose=1)
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 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.TimeDomain(0, 1)
ic = dde.IC(geom, func, lambda _, on_initial: on_initial)
quad_deg = 16
data = dde.data.IDE(geom, ide, ic, 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 solve_heat_with_dl(lightweight=False):
geom = dde.geometry.Rectangle(xmin=[0, 0], xmax=[1, 1])
timedomain = dde.geometry.TimeDomain(0, 2)
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,
ic_bcs=[ic, bc],
num_domain=500,
num_boundary=300,
num_initial=200,
num_test=10000,
solution=_func,
)
# inputs (2D Heat Eqn) - t, x, y
# output (solution) - u(t, x, y)
layer_size = [3] + [128] + [256] + [521] + [256] + [128] + [1]
activation = 'tanh'
initializer = 'Glorot uniform'
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
# train model
model.compile('adam', lr=0.001, metrics=['l2 relative error'])
losshistory, train_state = model.train(epochs=10000)
if not lightweight:
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
save_dynamic_contours_from_model(model, 1, 1, 2, 100, 'heat2d')
return model
def train_source_model(d):
path = Path('Neural_Networks', 'd_{}'.format(d))
if path.exists() and path.is_dir():
shutil.rmtree(path)
os.mkdir(path)
def pde(x, u):
u_vel, v_vel, w_vel, p = u[:, 0:1], u[:, 1:2], u[:, 2:3], u[:, 3:4]
u_vel_x = dde.grad.jacobian(u, x, i=0, j=0)
u_vel_y = dde.grad.jacobian(u, x, i=0, j=1)
u_vel_z = dde.grad.jacobian(u, x, i=0, j=2)
u_vel_t = dde.grad.jacobian(u, x, i=0, j=3)
u_vel_xx = dde.grad.hessian(u, x, component=0, i=0, j=0)
u_vel_yy = dde.grad.hessian(u, x, component=0, i=1, j=1)
u_vel_zz = dde.grad.hessian(u, x, component=0, i=2, j=2)
v_vel_x = dde.grad.jacobian(u, x, i=1, j=0)
v_vel_y = dde.grad.jacobian(u, x, i=1, j=1)
v_vel_z = dde.grad.jacobian(u, x, i=1, j=2)
v_vel_t = dde.grad.jacobian(u, x, i=1, j=3)
v_vel_xx = dde.grad.hessian(u, x, component=1, i=0, j=0)
v_vel_yy = dde.grad.hessian(u, x, component=1, i=1, j=1)
v_vel_zz = dde.grad.hessian(u, x, component=1, i=2, j=2)
w_vel_x = dde.grad.jacobian(u, x, i=2, j=0)
w_vel_y = dde.grad.jacobian(u, x, i=2, j=1)
w_vel_z = dde.grad.jacobian(u, x, i=2, j=2)
w_vel_t = dde.grad.jacobian(u, x, i=2, j=3)
w_vel_xx = dde.grad.hessian(u, x, component=2, i=0, j=0)
w_vel_yy = dde.grad.hessian(u, x, component=2, i=1, j=1)
w_vel_zz = dde.grad.hessian(u, x, component=2, i=2, j=2)
p_x = dde.grad.jacobian(u, x, i=3, j=0)
p_y = dde.grad.jacobian(u, x, i=3, j=1)
p_z = dde.grad.jacobian(u, x, i=3, j=2)
momentum_x = u_vel_t + (
u_vel * u_vel_x + v_vel * u_vel_y +
w_vel * u_vel_z) + p_x - 1 / Re * (u_vel_xx + u_vel_yy + u_vel_zz)
momentum_y = v_vel_t + (
u_vel * v_vel_x + v_vel * v_vel_y +
w_vel * v_vel_z) + p_y - 1 / Re * (v_vel_xx + v_vel_yy + v_vel_zz)
momentum_z = w_vel_t + (
u_vel * w_vel_x + v_vel * w_vel_y +
w_vel * w_vel_z) + p_z - 1 / Re * (w_vel_xx + w_vel_yy + w_vel_zz)
continuity = u_vel_x + v_vel_y + w_vel_z
return [momentum_x, momentum_y, momentum_z, continuity]
def u_func(x):
return -a * (
np.exp(a * x[:, 0:1]) * np.sin(a * x[:, 1:2] + d * x[:, 2:3]) +
np.exp(a * x[:, 2:3]) *
np.cos(a * x[:, 0:1] + d * x[:, 1:2])) * np.exp(-d**2 * x[:, 3:4])
def v_func(x):
return -a * (
np.exp(a * x[:, 1:2]) * np.sin(a * x[:, 2:3] + d * x[:, 0:1]) +
np.exp(a * x[:, 0:1]) *
np.cos(a * x[:, 1:2] + d * x[:, 2:3])) * np.exp(-d**2 * x[:, 3:4])
def w_func(x):
return -a * (
np.exp(a * x[:, 2:3]) * np.sin(a * x[:, 0:1] + d * x[:, 1:2]) +
np.exp(a * x[:, 1:2]) *
np.cos(a * x[:, 2:3] + d * x[:, 0:1])) * np.exp(-d**2 * x[:, 3:4])
def p_func(x):
return -1 / 2 * a**2 * (np.exp(2 * a * x[:, 0:1]) + np.exp(
2 * a * x[:, 0:1]) + np.exp(2 * a * x[:, 2:3]) +
2 * np.exp(a * x[:, 0:1] + d * x[:, 1:2]) *
np.cos(a * x[:, 2:3] + d * x[:, 0:1]) *
np.exp(a * (x[:, 1:2] + x[:, 2:3])) +
2 * np.exp(a * x[:, 1:2] + d * x[:, 2:3]) *
np.cos(a * x[:, 0:1] + d * x[:, 1:2]) *
np.exp(a * (x[:, 2:3] + x[:, 0:1])) +
2 * np.exp(a * x[:, 2:3] + d * x[:, 0:1]) *
np.cos(a * x[:, 1:2] + d * x[:, 2:3]) *
np.exp(a * (x[:, 0:1] + x[:, 1:2]))) * np.exp(
-2 * d**2 * x[:, 3:4])
spatial_domain = dde.geometry.geometry_3d.Cuboid(
xmin=[x_min, y_min, z_min], xmax=[x_max, y_max, z_max])
temporal_domain = dde.geometry.TimeDomain(t_min, t_max)
spatio_temporal_domain = dde.geometry.GeometryXTime(
spatial_domain, temporal_domain)
boundary_condition_u = dde.DirichletBC(spatio_temporal_domain,
u_func,
lambda _, on_boundary: on_boundary,
component=0)
boundary_condition_v = dde.DirichletBC(spatio_temporal_domain,
v_func,
lambda _, on_boundary: on_boundary,
component=1)
boundary_condition_w = dde.DirichletBC(spatio_temporal_domain,
w_func,
lambda _, on_boundary: on_boundary,
component=2)
#boundary_condition_p = dde.DirichletBC(spatio_temporal_domain, p_func, lambda _, on_boundary: on_boundary, component=3)
initial_condition_u = dde.IC(spatio_temporal_domain,
u_func,
lambda _, on_initial: on_initial,
component=0)
initial_condition_v = dde.IC(spatio_temporal_domain,
v_func,
lambda _, on_initial: on_initial,
component=1)
initial_condition_w = dde.IC(spatio_temporal_domain,
w_func,
lambda _, on_initial: on_initial,
component=2)
#initial_condition_p = dde.IC(spatio_temporal_domain, p_func, lambda _, on_initial: on_initial, component = 3)
data = dde.data.TimePDE(
spatio_temporal_domain,
pde, [
boundary_condition_u, boundary_condition_v, boundary_condition_w,
initial_condition_u, initial_condition_v, initial_condition_w
],
num_domain=domain_points,
num_boundary=boundary_points,
num_initial=initial_points,
num_test=test_points)
model_name = 'Neural_Networks/d_{}/Beltrami_Flow_Source_Model_d_{}'.format(
d, d)
net = dde.maps.FNN([4] + hidden_layers * [hidden_units] + [4], "tanh",
"Glorot normal")
model = dde.Model(data, net)
start = time.time()
model.compile("adam",
lr=learning_rate,
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
model.train(epochs=number_of_epochs)
model.compile("L-BFGS-B",
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
losshistory, train_state = model.train(model_save_path=model_name)
end = time.time()
length = end - start
x, y, z = np.meshgrid(
np.linspace(x_min, x_max, test_points_per_dimension),
np.linspace(y_min, y_max, test_points_per_dimension),
np.linspace(z_min, z_max, test_points_per_dimension),
)
X = np.vstack((np.ravel(x), np.ravel(y), np.ravel(z))).T
t_0 = t_min * np.ones(test_points).reshape(test_points, 1)
t_1 = t_max * np.ones(test_points).reshape(test_points, 1)
X_0 = np.hstack((X, t_0))
X_1 = np.hstack((X, t_1))
output_0 = model.predict(X_0)
output_1 = model.predict(X_1)
u_pred_0 = output_0[:, 0].reshape(-1)
v_pred_0 = output_0[:, 1].reshape(-1)
w_pred_0 = output_0[:, 2].reshape(-1)
p_pred_0 = output_0[:, 3].reshape(-1)
u_exact_0 = u_func(X_0).reshape(-1)
v_exact_0 = v_func(X_0).reshape(-1)
w_exact_0 = w_func(X_0).reshape(-1)
p_exact_0 = p_func(X_0).reshape(-1)
u_pred_1 = output_1[:, 0].reshape(-1)
v_pred_1 = output_1[:, 1].reshape(-1)
w_pred_1 = output_1[:, 2].reshape(-1)
p_pred_1 = output_1[:, 3].reshape(-1)
u_exact_1 = u_func(X_1).reshape(-1)
v_exact_1 = v_func(X_1).reshape(-1)
w_exact_1 = w_func(X_1).reshape(-1)
p_exact_1 = p_func(X_1).reshape(-1)
f_0 = model.predict(X_0, operator=pde)
f_1 = model.predict(X_1, operator=pde)
l2_difference_u_0 = dde.metrics.l2_relative_error(u_exact_0, u_pred_0)
l2_difference_v_0 = dde.metrics.l2_relative_error(v_exact_0, v_pred_0)
l2_difference_w_0 = dde.metrics.l2_relative_error(w_exact_0, w_pred_0)
l2_difference_p_0 = dde.metrics.l2_relative_error(p_exact_0, p_pred_0)
residual_0 = np.mean(np.absolute(f_0))
l2_difference_u_1 = dde.metrics.l2_relative_error(u_exact_1, u_pred_1)
l2_difference_v_1 = dde.metrics.l2_relative_error(v_exact_1, v_pred_1)
l2_difference_w_1 = dde.metrics.l2_relative_error(w_exact_1, w_pred_1)
l2_difference_p_1 = dde.metrics.l2_relative_error(p_exact_1, p_pred_1)
residual_1 = np.mean(np.absolute(f_1))
final_epochs = train_state.epoch
return l2_difference_u_0, l2_difference_v_0, l2_difference_w_0, l2_difference_p_0, residual_0, l2_difference_u_1, l2_difference_v_1, l2_difference_w_1, l2_difference_p_1, residual_1, length, final_epochs
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)
'''
check rho0
for i in range(0,len(ob_y[1,:])):
for j in range(0,len(x0[1,:])):
plt.plot(x0[:,j],ob_y[:,i],'.')
plt.show()
'''
timedomain = dde.geometry.TimeDomain(0, tmax)
geom = dde.geometry.GeometryXTime(geom, timedomain)
# test points setting
# x0 = np.random.random([1000,C.N+1])
# x0[:,0:C.N] = 2 * Qmax*(x0[:,0:C.N]-0.5)
# ob_y = initialState(x0)
# ptset = dde.bc.PointSet(x0)
# inside = lambda x, _: ptset.inside(x)
# 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=2000,
num_boundary=0,
num_initial=500,
num_test=None)
# settings for model
layer_size = [C.N + 1] + [100] * 5 + [(C.N + 1)**2]
activation = "tanh"
initializer = "Glorot uniform"
save_model_dir = os.path.join(os.getcwd(), 'Model_save', 'test_rho0_0102')
if not os.path.isdir(save_model_dir):
os.mkdir(save_model_dir)
save_model_name = os.path.join(save_model_dir, 'model_save')
load_epoch = '15000'
load_model_name = save_model_name + '-' + load_epoch
Callfcn = dde.callbacks.ModelCheckpoint(save_model_name,
verbose=1,
save_better_only=True,
period=10000)
# initialize model
net = dde.maps.FNN(layer_size, activation, initializer)
net.apply_output_transform(boundary_condition)
model = dde.Model(data, net)
model.compile("adam", lr=0.001)
model.restore(load_model_name)
losshistory, train_state = model.train(epochs=60000,
callbacks=[Callfcn],
model_save_path=save_model_name)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
# Definition of the Dirichlet BCs
bc_u_no_slip = dde.DirichletBC(geomtime,
zero_velocity,
zero_boundary,
component=0)
bc_v_no_slip = dde.DirichletBC(geomtime,
zero_velocity,
zero_boundary,
component=1)
bc_u_cylinder = dde.DirichletBC(geomtime, zero_velocity, cylinder, component=0)
bc_v_cylinder = dde.DirichletBC(geomtime, zero_velocity, cylinder, component=1)
bc_u_inlet = dde.DirichletBC(geomtime, inlet_velocity, inlet, component=0)
bc_v_inlet = dde.DirichletBC(geomtime, inlet_velocity, inlet, component=1)
bc_p = dde.DirichletBC(geomtime, zero_pressure, outlet, component=2)
ic = dde.IC(geomtime, zero_velocity, on_initial)
bc = [
bc_u_no_slip, bc_v_no_slip, bc_u_cylinder, bc_v_cylinder, bc_u_inlet,
bc_v_inlet, bc_p, ic
]
# Domain sampling and data generation
data = dde.data.TimePDE(geomtime,
pde,
bc,
num_domain=Nd,
num_boundary=Nb,
num_initial=Nt)
# Definition of the FFNN architecture
def main():
data = get_covid_data()
#sir params
beta = tf.Variable(.0001)
gamma = tf.Variable(.0002)
N = data[1][0][0] + data[1][0][1] #population
initial_infected = data[1][0][1]
initial_removed = data[1][0][2]
#x is time, y is odes
def odes(x, y):
s, i, r = y[:, 0:1], y[:, 1:2], y[:, 2:3]
ds_x = tf.gradients(s, x)[0]
di_x = tf.gradients(i, x)[0]
dr_x = tf.gradients(r, x)[0]
de1 = ds_x + abs(beta) * i * s / N
de2 = di_x - abs(beta) * i * s / N + (abs(gamma) * i)
de3 = dr_x - abs(gamma) * i
return [de1, de2, de3]
def boundary(_, on_initial):
return on_initial
def func(x):
return np.hstack(
(np.zeros(x.shape), np.zeros(x.shape), np.zeros(x.shape)))
geom = dde.geometry.TimeDomain(0, len(data[0]))
ic1 = dde.IC(geom,
lambda x:
(N - initial_infected - initial_removed) * np.ones(x.shape),
boundary,
component=0)
ic2 = dde.IC(geom,
lambda x: initial_infected * np.ones(x.shape),
boundary,
component=1)
ic3 = dde.IC(geom,
lambda x: initial_removed * np.ones(x.shape),
boundary,
component=2)
observe_t, observe_y = data
ptset = dde.bc.PointSet(observe_t)
inside = lambda x, _: ptset.inside(x)
observe_y0 = dde.DirichletBC(geom,
ptset.values_to_func(observe_y[:, 0:1]),
inside,
component=0)
observe_y1 = dde.DirichletBC(geom,
ptset.values_to_func(observe_y[:, 1:2]),
inside,
component=1)
observe_y2 = dde.DirichletBC(geom,
ptset.values_to_func(observe_y[:, 2:3]),
inside,
component=2)
data = dde.data.PDE(geom,
odes, [ic2, ic3, observe_y1, observe_y2],
num_domain=300,
num_boundary=1,
anchors=observe_t)
net = dde.maps.FNN([1] + [40] * 4 + [3], 'tanh', 'Glorot uniform')
model = dde.Model(data, net)
model.compile("adam", lr=.01, loss='mse') # try 'log cosh'
variable = dde.callbacks.VariableValue([beta, gamma],
period=1000,
filename="variables.dat")
losshistory, train_state = model.train(epochs=10000, callbacks=[variable])
plot_best_state(train_state)
plot_loss_history(losshistory)
plt.show()
def visualize_solution(d, epochs_source):
def pde(x, u):
u_vel, v_vel, w_vel, p = u[:, 0:1], u[:, 1:2], u[:, 2:3], u[:, 3:4]
u_vel_x = dde.grad.jacobian(u, x, i=0, j=0)
u_vel_y = dde.grad.jacobian(u, x, i=0, j=1)
u_vel_z = dde.grad.jacobian(u, x, i=0, j=2)
u_vel_t = dde.grad.jacobian(u, x, i=0, j=3)
u_vel_xx = dde.grad.hessian(u, x, component=0, i=0, j=0)
u_vel_yy = dde.grad.hessian(u, x, component=0, i=1, j=1)
u_vel_zz = dde.grad.hessian(u, x, component=0, i=2, j=2)
v_vel_x = dde.grad.jacobian(u, x, i=1, j=0)
v_vel_y = dde.grad.jacobian(u, x, i=1, j=1)
v_vel_z = dde.grad.jacobian(u, x, i=1, j=2)
v_vel_t = dde.grad.jacobian(u, x, i=1, j=3)
v_vel_xx = dde.grad.hessian(u, x, component=1, i=0, j=0)
v_vel_yy = dde.grad.hessian(u, x, component=1, i=1, j=1)
v_vel_zz = dde.grad.hessian(u, x, component=1, i=2, j=2)
w_vel_x = dde.grad.jacobian(u, x, i=2, j=0)
w_vel_y = dde.grad.jacobian(u, x, i=2, j=1)
w_vel_z = dde.grad.jacobian(u, x, i=2, j=2)
w_vel_t = dde.grad.jacobian(u, x, i=2, j=3)
w_vel_xx = dde.grad.hessian(u, x, component=2, i=0, j=0)
w_vel_yy = dde.grad.hessian(u, x, component=2, i=1, j=1)
w_vel_zz = dde.grad.hessian(u, x, component=2, i=2, j=2)
p_x = dde.grad.jacobian(u, x, i=3, j=0)
p_y = dde.grad.jacobian(u, x, i=3, j=1)
p_z = dde.grad.jacobian(u, x, i=3, j=2)
momentum_x = u_vel_t + (
u_vel * u_vel_x + v_vel * u_vel_y +
w_vel * u_vel_z) + p_x - 1 / Re * (u_vel_xx + u_vel_yy + u_vel_zz)
momentum_y = v_vel_t + (
u_vel * v_vel_x + v_vel * v_vel_y +
w_vel * v_vel_z) + p_y - 1 / Re * (v_vel_xx + v_vel_yy + v_vel_zz)
momentum_z = w_vel_t + (
u_vel * w_vel_x + v_vel * w_vel_y +
w_vel * w_vel_z) + p_z - 1 / Re * (w_vel_xx + w_vel_yy + w_vel_zz)
continuity = u_vel_x + v_vel_y + w_vel_z
return [momentum_x, momentum_y, momentum_z, continuity]
def u_func(x):
return -a * (
np.exp(a * x[:, 0:1]) * np.sin(a * x[:, 1:2] + d * x[:, 2:3]) +
np.exp(a * x[:, 2:3]) *
np.cos(a * x[:, 0:1] + d * x[:, 1:2])) * np.exp(-d**2 * x[:, 3:4])
def v_func(x):
return -a * (
np.exp(a * x[:, 1:2]) * np.sin(a * x[:, 2:3] + d * x[:, 0:1]) +
np.exp(a * x[:, 0:1]) *
np.cos(a * x[:, 1:2] + d * x[:, 2:3])) * np.exp(-d**2 * x[:, 3:4])
def w_func(x):
return -a * (
np.exp(a * x[:, 2:3]) * np.sin(a * x[:, 0:1] + d * x[:, 1:2]) +
np.exp(a * x[:, 1:2]) *
np.cos(a * x[:, 2:3] + d * x[:, 0:1])) * np.exp(-d**2 * x[:, 3:4])
def p_func(x):
return -1 / 2 * a**2 * (np.exp(2 * a * x[:, 0:1]) + np.exp(
2 * a * x[:, 0:1]) + np.exp(2 * a * x[:, 2:3]) +
2 * np.exp(a * x[:, 0:1] + d * x[:, 1:2]) *
np.cos(a * x[:, 2:3] + d * x[:, 0:1]) *
np.exp(a * (x[:, 1:2] + x[:, 2:3])) +
2 * np.exp(a * x[:, 1:2] + d * x[:, 2:3]) *
np.cos(a * x[:, 0:1] + d * x[:, 1:2]) *
np.exp(a * (x[:, 2:3] + x[:, 0:1])) +
2 * np.exp(a * x[:, 2:3] + d * x[:, 0:1]) *
np.cos(a * x[:, 1:2] + d * x[:, 2:3]) *
np.exp(a * (x[:, 0:1] + x[:, 1:2]))) * np.exp(
-2 * d**2 * x[:, 3:4])
spatial_domain = dde.geometry.geometry_3d.Cuboid(
xmin=[x_min, y_min, z_min], xmax=[x_max, y_max, z_max])
temporal_domain = dde.geometry.TimeDomain(t_min, t_max)
spatio_temporal_domain = dde.geometry.GeometryXTime(
spatial_domain, temporal_domain)
boundary_condition_u = dde.DirichletBC(spatio_temporal_domain,
u_func,
lambda _, on_boundary: on_boundary,
component=0)
boundary_condition_v = dde.DirichletBC(spatio_temporal_domain,
v_func,
lambda _, on_boundary: on_boundary,
component=1)
boundary_condition_w = dde.DirichletBC(spatio_temporal_domain,
w_func,
lambda _, on_boundary: on_boundary,
component=2)
#boundary_condition_p = dde.DirichletBC(spatio_temporal_domain, p_func, lambda _, on_boundary: on_boundary, component=3)
initial_condition_u = dde.IC(spatio_temporal_domain,
u_func,
lambda _, on_initial: on_initial,
component=0)
initial_condition_v = dde.IC(spatio_temporal_domain,
v_func,
lambda _, on_initial: on_initial,
component=1)
initial_condition_w = dde.IC(spatio_temporal_domain,
w_func,
lambda _, on_initial: on_initial,
component=2)
#initial_condition_p = dde.IC(spatio_temporal_domain, p_func, lambda _, on_initial: on_initial, component = 3)
data = dde.data.TimePDE(
spatio_temporal_domain,
pde, [
boundary_condition_u, boundary_condition_v, boundary_condition_w,
initial_condition_u, initial_condition_v, initial_condition_w
],
num_domain=domain_points,
num_boundary=boundary_points,
num_initial=initial_points,
num_test=test_points)
net = dde.maps.FNN([4] + hidden_layers * [hidden_units] + [4], "tanh",
"Glorot normal")
model = dde.Model(data, net)
# Compute reference solution
x, y, z = np.meshgrid(
np.linspace(x_min, x_max, test_points_per_dimension),
np.linspace(y_min, y_max, test_points_per_dimension),
np.linspace(z_min, z_max, test_points_per_dimension),
)
X = np.vstack((np.ravel(x), np.ravel(y), np.ravel(z))).T
t_0 = t_min * np.ones(test_points).reshape(test_points, 1)
t_1 = t_max * np.ones(test_points).reshape(test_points, 1)
X_0 = np.hstack((X, t_0))
X_1 = np.hstack((X, t_1))
# Reload model and make predictions
model_name = 'Neural_Networks/d_{}/Beltrami_Flow_Source_Model_d_{}-{}'.format(
d, d, epochs_source)
model.compile("adam", lr=learning_rate)
model.train(model_restore_path=model_name, epochs=0)
output_0 = model.predict(X_0)
output_1 = model.predict(X_1)
u_pred_0 = output_0[:, 0].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
v_pred_0 = output_0[:, 1].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
w_pred_0 = output_0[:, 2].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
p_pred_0 = output_0[:, 3].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
u_pred_1 = output_1[:, 0].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
v_pred_1 = output_1[:, 1].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
w_pred_1 = output_1[:, 2].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
p_pred_1 = output_1[:, 3].reshape(test_points_per_dimension,
test_points_per_dimension,
test_points_per_dimension)
plt.figure(0)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], u_pred_0[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('u')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_u_d_{}_t_0.png'.format(d),
dpi=300)
plt.figure(1)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], u_pred_1[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('u')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_u_d_{}_t_1.png'.format(d),
dpi=300)
plt.figure(2)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], v_pred_0[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('v')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_v_d_{}_t_0.png'.format(d),
dpi=300)
plt.figure(3)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], v_pred_1[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('v')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_v_d_{}_t_1.png'.format(d),
dpi=300)
plt.figure(4)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], w_pred_0[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('w')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_w_d_{}_t_0.png'.format(d),
dpi=300)
plt.figure(5)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], w_pred_1[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('w')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_w_d_{}_t_1.png'.format(d),
dpi=300)
plt.figure(6)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], p_pred_0[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('p')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_p_d_{}_t_0.png'.format(d),
dpi=300)
plt.figure(7)
ax = plt.axes(projection="3d")
ax.plot_wireframe(x[:, :, z_index], y[:, :, z_index], p_pred_1[:, :,
z_index])
ax.set_xlabel('location x')
ax.set_ylabel('location y')
ax.set_zlabel('p')
plt.tight_layout()
plt.savefig('Predictions/Predicted_solution_p_d_{}_t_1.png'.format(d),
dpi=300)
return
def train_target_model(nus_source, epochs_source, nu_target,
optimization_strategy):
individual_similarities = np.zeros(len(nus_source))
for k in range(len(nus_source)):
individual_similarities[k] = np.linalg.norm(
nu_target - nus_source[k]) / np.linalg.norm(nu_target)
index = np.unravel_index(np.argmin(individual_similarities, axis=None),
individual_similarities.shape)[0]
similarity = individual_similarities[index]
closest_nu = nus_source[index]
file_name = 'Reference_Solutions/u_exact_nu_{}.mat'.format(nu_target)
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
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_target * 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)
start = time.time()
if optimization_strategy == 0:
print("L-BFGS (random)")
model.compile("L-BFGS-B")
model.train()
elif optimization_strategy == 1:
print("L-BFGS (smart)")
model_name_source = '../Source/Neural_Networks/nu_{}/Burger_Equation_Source_Model_nu_{}-{}'.format(
closest_nu, closest_nu, epochs_source[index])
model.compile("L-BFGS-B")
model.train(model_restore_path=model_name_source)
else:
print("Adam + L-BFGS (random)")
model.compile("adam", lr=learning_rate)
model.train(epochs=number_of_epochs)
model.compile("L-BFGS-B")
model.train()
end = time.time()
length = end - start
u_pred = model.predict(X).reshape(test_points_t, test_points_x)
f = model.predict(X, operator=pde)
residual = np.mean(np.absolute(f))
l2_difference = dde.metrics.l2_relative_error(u_exact, u_pred)
return l2_difference, residual, similarity, length
def check_x0():
"""
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)
# x0 = np.random.random([1000,C.N+1])
x0 = np.load('./rand_x0.npy')
xtest = np.random.random([10000, C.N + 1])
ytest = initialState(xtest)
ob_y = initialState(x0)
geom = dde.geometry.Hypercube(Xmin, Xmax)
# geom = dde.geometry.Interval(-1, 1)
'''
check rho0
for i in range(0,len(ob_y[1,:])):
for j in range(0,len(x0[1,:])):
plt.plot(x0[:,j],ob_y[:,i],'.')
plt.show()
'''
timedomain = dde.geometry.TimeDomain(0, tmax)
geom = dde.geometry.GeometryXTime(geom, timedomain)
# Initial conditions
ic = []
ic_nb = []
ptset = dde.bc.PointSet(x0)
inside = lambda x, _: ptset.inside(x)
for j in range(0, (C.N + 1)**2):
ic.append(
dde.IC(geom, lambda X: initialState(X, j), boundary, component=j))
ic.append(
dde.DirichletBC(geom,
ptset.values_to_func(ob_y[:, j:j + 1]),
inside,
component=j))
ic_nb.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_sle = dde.data.TimePDE(geom,
lambda x, y: SLE_q(x, y, C),
ic_nb,
num_domain=1000,
num_boundary=0,
num_initial=100,
num_test=None)
data = dde.data.TimePDE(geom,
lambda x, y: y * 0,
ic,
num_domain=0,
num_boundary=0,
num_initial=100,
anchors=x0,
num_test=None)
# lambda x,y: SLE_q(x,y,C),
# settings for model
# layer_size = [C.N+1] + [100] * 5 + [(C.N+1) **2]
layer_size = [C.N + 1] + [50] * 3 + [(C.N + 1)**2]
activation = "tanh"
initializer = "Glorot uniform"
save_model_dir = os.path.join(os.getcwd(), 'Model_save', 'test_rho0_0101')
if not os.path.isdir(save_model_dir):
os.mkdir(save_model_dir)
save_model_name = os.path.join(save_model_dir, 'model_test')
load_epoch = '65000'
load_model_name = save_model_name + '-' + load_epoch
Callfcn = dde.callbacks.ModelCheckpoint(save_model_name,
verbose=1,
save_better_only=True,
period=10000)
# initialize model
net = dde.maps.FNN(layer_size, activation, initializer)
net.apply_output_transform(boundary_condition)
model = dde.Model(data, net)
# load model
model.compile("adam", lr=0.001)
model.restore(load_model_name)
y_pre = model.predict(x0)
x1 = x0
x1[:, 1] = x1[:, 1] * 0
# x1[:,2]=x1[:,2]*0
y_pre2 = model.predict(x1)
# load_model_name = save_model_name + '-' + '5000'
# model.restore(load_model_name)
# y_pre = model.predict(x0)
for i in range(0, 9):
plt.subplot(3, 3, i + 1)
plt.plot(x0[:, 0], ob_y[:, i], '.b')
plt.plot(x0[:, 0], y_pre[:, i], '.r')
plt.plot(x1[:, 0], y_pre2[:, i], '.g')
plt.show()
def train_target_model(ds_source, epochs_source, d_target,
optimization_strategy):
individual_similarities = np.zeros(len(ds_source))
for k in range(len(ds_source)):
individual_similarities[k] = 1 - np.linalg.norm(
d_target - ds_source[k]) / np.linalg.norm(d_target)
index = np.unravel_index(np.argmax(individual_similarities, axis=None),
individual_similarities.shape)[0]
similarity = individual_similarities[index]
closest_d = ds_source[index]
def pde(x, u):
u_vel, v_vel, w_vel, p = u[:, 0:1], u[:, 1:2], u[:, 2:3], u[:, 3:4]
u_vel_x = dde.grad.jacobian(u, x, i=0, j=0)
u_vel_y = dde.grad.jacobian(u, x, i=0, j=1)
u_vel_z = dde.grad.jacobian(u, x, i=0, j=2)
u_vel_t = dde.grad.jacobian(u, x, i=0, j=3)
u_vel_xx = dde.grad.hessian(u, x, component=0, i=0, j=0)
u_vel_yy = dde.grad.hessian(u, x, component=0, i=1, j=1)
u_vel_zz = dde.grad.hessian(u, x, component=0, i=2, j=2)
v_vel_x = dde.grad.jacobian(u, x, i=1, j=0)
v_vel_y = dde.grad.jacobian(u, x, i=1, j=1)
v_vel_z = dde.grad.jacobian(u, x, i=1, j=2)
v_vel_t = dde.grad.jacobian(u, x, i=1, j=3)
v_vel_xx = dde.grad.hessian(u, x, component=1, i=0, j=0)
v_vel_yy = dde.grad.hessian(u, x, component=1, i=1, j=1)
v_vel_zz = dde.grad.hessian(u, x, component=1, i=2, j=2)
w_vel_x = dde.grad.jacobian(u, x, i=2, j=0)
w_vel_y = dde.grad.jacobian(u, x, i=2, j=1)
w_vel_z = dde.grad.jacobian(u, x, i=2, j=2)
w_vel_t = dde.grad.jacobian(u, x, i=2, j=3)
w_vel_xx = dde.grad.hessian(u, x, component=2, i=0, j=0)
w_vel_yy = dde.grad.hessian(u, x, component=2, i=1, j=1)
w_vel_zz = dde.grad.hessian(u, x, component=2, i=2, j=2)
p_x = dde.grad.jacobian(u, x, i=3, j=0)
p_y = dde.grad.jacobian(u, x, i=3, j=1)
p_z = dde.grad.jacobian(u, x, i=3, j=2)
momentum_x = u_vel_t + (
u_vel * u_vel_x + v_vel * u_vel_y +
w_vel * u_vel_z) + p_x - 1 / Re * (u_vel_xx + u_vel_yy + u_vel_zz)
momentum_y = v_vel_t + (
u_vel * v_vel_x + v_vel * v_vel_y +
w_vel * v_vel_z) + p_y - 1 / Re * (v_vel_xx + v_vel_yy + v_vel_zz)
momentum_z = w_vel_t + (
u_vel * w_vel_x + v_vel * w_vel_y +
w_vel * w_vel_z) + p_z - 1 / Re * (w_vel_xx + w_vel_yy + w_vel_zz)
continuity = u_vel_x + v_vel_y + w_vel_z
return [momentum_x, momentum_y, momentum_z, continuity]
def u_func(x):
return -a * (np.exp(a * x[:, 0:1]) *
np.sin(a * x[:, 1:2] + d_target * x[:, 2:3]) +
np.exp(a * x[:, 2:3]) *
np.cos(a * x[:, 0:1] + d_target * x[:, 1:2])) * np.exp(
-d_target**2 * x[:, 3:4])
def v_func(x):
return -a * (np.exp(a * x[:, 1:2]) *
np.sin(a * x[:, 2:3] + d_target * x[:, 0:1]) +
np.exp(a * x[:, 0:1]) *
np.cos(a * x[:, 1:2] + d_target * x[:, 2:3])) * np.exp(
-d_target**2 * x[:, 3:4])
def w_func(x):
return -a * (np.exp(a * x[:, 2:3]) *
np.sin(a * x[:, 0:1] + d_target * x[:, 1:2]) +
np.exp(a * x[:, 1:2]) *
np.cos(a * x[:, 2:3] + d_target * x[:, 0:1])) * np.exp(
-d_target**2 * x[:, 3:4])
def p_func(x):
return -1 / 2 * a**2 * (
np.exp(2 * a * x[:, 0:1]) + np.exp(2 * a * x[:, 0:1]) +
np.exp(2 * a * x[:, 2:3]) +
2 * np.exp(a * x[:, 0:1] + d_target * x[:, 1:2]) *
np.cos(a * x[:, 2:3] + d_target * x[:, 0:1]) *
np.exp(a * (x[:, 1:2] + x[:, 2:3])) +
2 * np.exp(a * x[:, 1:2] + d_target * x[:, 2:3]) *
np.cos(a * x[:, 0:1] + d_target * x[:, 1:2]) *
np.exp(a * (x[:, 2:3] + x[:, 0:1])) +
2 * np.exp(a * x[:, 2:3] + d_target * x[:, 0:1]) *
np.cos(a * x[:, 1:2] + d_target * x[:, 2:3]) *
np.exp(a * (x[:, 0:1] + x[:, 1:2]))) * np.exp(
-2 * d_target**2 * x[:, 3:4])
spatial_domain = dde.geometry.geometry_3d.Cuboid(
xmin=[x_min, y_min, z_min], xmax=[x_max, y_max, z_max])
temporal_domain = dde.geometry.TimeDomain(t_min, t_max)
spatio_temporal_domain = dde.geometry.GeometryXTime(
spatial_domain, temporal_domain)
boundary_condition_u = dde.DirichletBC(spatio_temporal_domain,
u_func,
lambda _, on_boundary: on_boundary,
component=0)
boundary_condition_v = dde.DirichletBC(spatio_temporal_domain,
v_func,
lambda _, on_boundary: on_boundary,
component=1)
boundary_condition_w = dde.DirichletBC(spatio_temporal_domain,
w_func,
lambda _, on_boundary: on_boundary,
component=2)
#boundary_condition_p = dde.DirichletBC(spatio_temporal_domain, p_func, lambda _, on_boundary: on_boundary, component=3)
initial_condition_u = dde.IC(spatio_temporal_domain,
u_func,
lambda _, on_initial: on_initial,
component=0)
initial_condition_v = dde.IC(spatio_temporal_domain,
v_func,
lambda _, on_initial: on_initial,
component=1)
initial_condition_w = dde.IC(spatio_temporal_domain,
w_func,
lambda _, on_initial: on_initial,
component=2)
#initial_condition_p = dde.IC(spatio_temporal_domain, p_func, lambda _, on_initial: on_initial, component = 3)
data = dde.data.TimePDE(
spatio_temporal_domain,
pde, [
boundary_condition_u, boundary_condition_v, boundary_condition_w,
initial_condition_u, initial_condition_v, initial_condition_w
],
num_domain=domain_points,
num_boundary=boundary_points,
num_initial=initial_points,
num_test=test_points)
net = dde.maps.FNN([4] + hidden_layers * [hidden_units] + [4], "tanh",
"Glorot normal")
model = dde.Model(data, net)
start = time.time()
if optimization_strategy == 0:
print("L-BFGS (random)")
model.compile("L-BFGS-B",
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
model.train()
elif optimization_strategy == 1:
print("L-BFGS (smart)")
model_name_source = '../Source/Neural_Networks/d_{}/Beltrami_Flow_Source_Model_d_{}-{}'.format(
closest_d, closest_d, epochs_source[index])
model.compile("L-BFGS-B",
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
model.train(model_restore_path=model_name_source)
else:
print("Adam + L-BFGS (random)")
model.compile("adam",
lr=learning_rate,
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
model.train(epochs=number_of_epochs)
model.compile("L-BFGS-B",
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
model.train()
end = time.time()
length = end - start
x, y, z = np.meshgrid(
np.linspace(x_min, x_max, test_points_per_dimension),
np.linspace(y_min, y_max, test_points_per_dimension),
np.linspace(z_min, z_max, test_points_per_dimension),
)
X = np.vstack((np.ravel(x), np.ravel(y), np.ravel(z))).T
t_0 = t_min * np.ones(test_points).reshape(test_points, 1)
t_1 = t_max * np.ones(test_points).reshape(test_points, 1)
X_0 = np.hstack((X, t_0))
X_1 = np.hstack((X, t_1))
output_0 = model.predict(X_0)
output_1 = model.predict(X_1)
u_pred_0 = output_0[:, 0].reshape(-1)
v_pred_0 = output_0[:, 1].reshape(-1)
w_pred_0 = output_0[:, 2].reshape(-1)
p_pred_0 = output_0[:, 3].reshape(-1)
u_exact_0 = u_func(X_0).reshape(-1)
v_exact_0 = v_func(X_0).reshape(-1)
w_exact_0 = w_func(X_0).reshape(-1)
p_exact_0 = p_func(X_0).reshape(-1)
u_pred_1 = output_1[:, 0].reshape(-1)
v_pred_1 = output_1[:, 1].reshape(-1)
w_pred_1 = output_1[:, 2].reshape(-1)
p_pred_1 = output_1[:, 3].reshape(-1)
u_exact_1 = u_func(X_1).reshape(-1)
v_exact_1 = v_func(X_1).reshape(-1)
w_exact_1 = w_func(X_1).reshape(-1)
p_exact_1 = p_func(X_1).reshape(-1)
f_0 = model.predict(X_0, operator=pde)
f_1 = model.predict(X_1, operator=pde)
l2_difference_u_0 = dde.metrics.l2_relative_error(u_exact_0, u_pred_0)
l2_difference_v_0 = dde.metrics.l2_relative_error(v_exact_0, v_pred_0)
l2_difference_w_0 = dde.metrics.l2_relative_error(w_exact_0, w_pred_0)
l2_difference_p_0 = dde.metrics.l2_relative_error(p_exact_0, p_pred_0)
residual_0 = np.mean(np.absolute(f_0))
l2_difference_u_1 = dde.metrics.l2_relative_error(u_exact_1, u_pred_1)
l2_difference_v_1 = dde.metrics.l2_relative_error(v_exact_1, v_pred_1)
l2_difference_w_1 = dde.metrics.l2_relative_error(w_exact_1, w_pred_1)
l2_difference_p_1 = dde.metrics.l2_relative_error(p_exact_1, p_pred_1)
residual_1 = np.mean(np.absolute(f_1))
return l2_difference_u_0, l2_difference_v_0, l2_difference_w_0, l2_difference_p_0, residual_0, l2_difference_u_1, l2_difference_v_1, l2_difference_w_1, l2_difference_p_1, residual_1, similarity, length
return [dy1_x - y2, dy2_x + y1]
def boundary(_, on_initial):
return on_initial
def func(x):
"""
y1 = sin(x)
y2 = cos(x)
"""
return np.hstack((np.sin(x), np.cos(x)))
geom = dde.geometry.TimeDomain(0, 10)
ic1 = dde.IC(geom, np.sin, boundary, component=0)
ic2 = dde.IC(geom, np.cos, boundary, component=1)
data = dde.data.PDE(geom, ode_system, [ic1, ic2], 35, 2, solution=func, num_test=100)
layer_size = [1] + [50] * 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=20000)
dde.saveplot(losshistory, train_state, issave=True, isplot=True)
def main():
# For starters, define some (dimensionless) parameters
a = 1
d = 1
Re = 1
# Introduce the span of the geometric domain
dx = 2
dy = 0.3
dz = 0.5
dt = 2e-3
def pde(x, u):
u_vel, v_vel, w_vel, p = u[:, 0:1], u[:, 1:2], u[:, 2:3], u[:, 3:4]
# Define the (partial) derivatives of the variables in a way that tensorflow understands
# We query corresponding entries of the jacobian and hessian and map them to new variables
# u
# First derivatives
u_vel_x = dde.grad.jacobian(u, x, i=0, j=0)
u_vel_y = dde.grad.jacobian(u, x, i=0, j=1)
u_vel_z = dde.grad.jacobian(u, x, i=0, j=2)
u_vel_t = dde.grad.jacobian(u, x, i=0, j=3)
# Second derivatives
u_vel_xx = dde.grad.hessian(u, x, component=0, i=0, j=0)
u_vel_yy = dde.grad.hessian(u, x, component=0, i=1, j=1)
u_vel_zz = dde.grad.hessian(u, x, component=0, i=2, j=2)
# v
# First derivatives
v_vel_x = dde.grad.jacobian(u, x, i=1, j=0)
v_vel_y = dde.grad.jacobian(u, x, i=1, j=1)
v_vel_z = dde.grad.jacobian(u, x, i=1, j=2)
v_vel_t = dde.grad.jacobian(u, x, i=1, j=3)
# Second derivatives
v_vel_xx = dde.grad.hessian(u, x, component=1, i=0, j=0)
v_vel_yy = dde.grad.hessian(u, x, component=1, i=1, j=1)
v_vel_zz = dde.grad.hessian(u, x, component=1, i=2, j=2)
# w
# First derivatives
w_vel_x = dde.grad.jacobian(u, x, i=2, j=0)
w_vel_y = dde.grad.jacobian(u, x, i=2, j=1)
w_vel_z = dde.grad.jacobian(u, x, i=2, j=2)
w_vel_t = dde.grad.jacobian(u, x, i=2, j=3)
# Second derivatives
w_vel_xx = dde.grad.hessian(u, x, component=2, i=0, j=0)
w_vel_yy = dde.grad.hessian(u, x, component=2, i=1, j=1)
w_vel_zz = dde.grad.hessian(u, x, component=2, i=2, j=2)
# pressure
p_x = dde.grad.jacobian(u, x, i=3, j=0)
p_y = dde.grad.jacobian(u, x, i=3, j=1)
p_z = dde.grad.jacobian(u, x, i=3, j=2)
# Assemble the momentum equations
momentum_x = (u_vel_t +
(u_vel * u_vel_x + v_vel * u_vel_y + w_vel * u_vel_z) +
p_x - 1 / Re * (u_vel_xx + u_vel_yy + u_vel_zz))
momentum_y = (v_vel_t +
(u_vel * v_vel_x + v_vel * v_vel_y + w_vel * v_vel_z) +
p_y - 1 / Re * (v_vel_xx + v_vel_yy + v_vel_zz))
momentum_z = (w_vel_t +
(u_vel * w_vel_x + v_vel * w_vel_y + w_vel * w_vel_z) +
p_z - 1 / Re * (w_vel_xx + w_vel_yy + w_vel_zz))
# Assemble continuity equation
continuity = u_vel_x + v_vel_y + w_vel_z
return [momentum_x, momentum_y, momentum_z, continuity]
# Make a simple cubic domain
spatial_domain = dde.geometry.Cuboid(xmin=[0, 0, 0], xmax=[dx, dy, dz])
# Specify the time domain with end time: 2 mm / (1000 mm/s) = 0.002 s
temporal_domain = dde.geometry.TimeDomain(0, dt)
spatio_temporal_domain = dde.geometry.GeometryXTime(
spatial_domain, temporal_domain)
# Define functions for the boundary conditions
def u_func(x):
return (
-a *
(np.exp(a * x[:, 0:1]) * np.sin(a * x[:, 1:2] + d * x[:, 2:3]) +
np.exp(a * x[:, 2:3]) * np.cos(a * x[:, 0:1] + d * x[:, 1:2])) *
np.exp(-(d**2) * x[:, 3:4]))
def v_func(x):
return (
-a *
(np.exp(a * x[:, 1:2]) * np.sin(a * x[:, 2:3] + d * x[:, 0:1]) +
np.exp(a * x[:, 0:1]) * np.cos(a * x[:, 1:2] + d * x[:, 2:3])) *
np.exp(-(d**2) * x[:, 3:4]))
def w_func(x):
return (
-a *
(np.exp(a * x[:, 2:3]) * np.sin(a * x[:, 0:1] + d * x[:, 1:2]) +
np.exp(a * x[:, 1:2]) * np.cos(a * x[:, 2:3] + d * x[:, 0:1])) *
np.exp(-(d**2) * x[:, 3:4]))
def p_func(x):
return (-0.5 * a**2 *
(np.exp(2 * a * x[:, 0:1]) + np.exp(2 * a * x[:, 0:1]) +
np.exp(2 * a * x[:, 2:3]) +
2 * np.exp(a * x[:, 0:1] + d * x[:, 1:2]) *
np.cos(a * x[:, 2:3] + d * x[:, 0:1]) *
np.exp(a * (x[:, 1:2] + x[:, 2:3])) +
2 * np.exp(a * x[:, 1:2] + d * x[:, 2:3]) *
np.cos(a * x[:, 0:1] + d * x[:, 1:2]) *
np.exp(a * (x[:, 2:3] + x[:, 0:1])) +
2 * np.exp(a * x[:, 2:3] + d * x[:, 0:1]) *
np.cos(a * x[:, 1:2] + d * x[:, 2:3]) *
np.exp(a * (x[:, 0:1] + x[:, 1:2]))) *
np.exp(-2 * d**2 * x[:, 3:4]))
# Define the boundary conditions
boundary_condition_u = dde.DirichletBC(spatio_temporal_domain,
u_func,
lambda _, on_boundary: on_boundary,
component=0)
boundary_condition_v = dde.DirichletBC(spatio_temporal_domain,
v_func,
lambda _, on_boundary: on_boundary,
component=1)
boundary_condition_w = dde.DirichletBC(spatio_temporal_domain,
w_func,
lambda _, on_boundary: on_boundary,
component=2)
# Define the initial conditions
initial_condition_u = dde.IC(spatio_temporal_domain,
u_func,
lambda _, on_initial: on_initial,
component=0)
initial_condition_v = dde.IC(spatio_temporal_domain,
v_func,
lambda _, on_initial: on_initial,
component=1)
initial_condition_w = dde.IC(spatio_temporal_domain,
w_func,
lambda _, on_initial: on_initial,
component=2)
# Define the data that is fed into the NN
data = dde.data.TimePDE(
spatio_temporal_domain,
pde,
[
boundary_condition_u,
boundary_condition_v,
boundary_condition_w,
initial_condition_u,
initial_condition_v,
initial_condition_w,
],
num_domain=50000,
num_boundary=5000,
num_initial=5000,
num_test=10000,
)
# Set up the NN architecture
# Using a Feedforward network (FNN) with 4 input and output layers and 4 hidden layers
# with 50 neurons each
net = dde.maps.FNN([4] + 4 * [50] + [4], "tanh", "Glorot normal")
model = dde.Model(data, net)
# Compile the model
# Use a learning rate of 1e-3 with an adam optimizer
model.compile("adam",
lr=1e-3,
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
# Train the model in 30000 epochs
model.train(epochs=30000)
model.compile("L-BFGS-B",
loss_weights=[1, 1, 1, 1, 100, 100, 100, 100, 100, 100])
losshistory, train_state = model.train()
# Generate the inference data
# Sample the domain
stepsize = 1e-3
x, y, z = np.meshgrid(
np.arange(0, dx, stepsize),
np.arange(0, dy, stepsize),
np.arange(0, dz, stepsize),
)
# Transform into a concatenated row vector
X = np.vstack((np.ravel(x), np.ravel(y), np.ravel(z))).T
# Create the time vectors
dim = (dx * dy * dz) / stepsize
t_0 = np.zeros(dim).reshape(dim, 1)
t_1 = np.zeros(dim).reshape(dim, 1)
t_1[:, 1] = dt
# Concatenate spatial and temporal vectors to spatiotemporal input arrays
X_0 = np.hstack(X, t_0)
X_1 = np.hstack(X, t_1)
# Calculate the inferred function values at the given points
output_0 = model.predict(X_0)
output_1 = model.predict(X_1)
# Get the separate fields from the output
# Remember, we gain four separate outputs
u_pred_0 = output_0[:, 0].reshape(-1)
v_pred_0 = output_0[:, 1].reshape(-1)
w_pred_0 = output_0[:, 2].reshape(-1)
p_pred_0 = output_0[:, 3].reshape(-1)
u_exact_0 = u_func(X_0).reshape(-1)
v_exact_0 = v_func(X_0).reshape(-1)
w_exact_0 = w_func(X_0).reshape(-1)
p_exact_0 = p_func(X_0).reshape(-1)
u_pred_1 = output_1[:, 0].reshape(-1)
v_pred_1 = output_1[:, 1].reshape(-1)
w_pred_1 = output_1[:, 2].reshape(-1)
p_pred_1 = output_1[:, 3].reshape(-1)
u_exact_1 = u_func(X_1).reshape(-1)
v_exact_1 = v_func(X_1).reshape(-1)
w_exact_1 = w_func(X_1).reshape(-1)
p_exact_1 = p_func(X_1).reshape(-1)
# Now, infer the function fitting the PDE
f_0 = model.predict(X_0, operator=pde)
f_1 = model.predict(X_1, operator=pde)
# Define the field-wise error metrics
l2_difference_u_0 = dde.metrics.l2_relative_error(u_exact_0, u_pred_0)
l2_difference_v_0 = dde.metrics.l2_relative_error(v_exact_0, v_pred_0)
l2_difference_w_0 = dde.metrics.l2_relative_error(w_exact_0, w_pred_0)
l2_difference_p_0 = dde.metrics.l2_realtive_error(p_exact_0, p_pred_0)
residual_0 = np.mean(np.absolute(f_0))
l2_difference_u_1 = dde.metrics.l2_relative_error(u_exact_1, u_pred_1)
l2_difference_v_1 = dde.metrics.l2_relative_error(v_exact_1, v_pred_1)
l2_difference_w_1 = dde.metrics.l2_relative_error(w_exact_1, w_pred_1)
l2_difference_p_1 = dde.metrics.l2_realtive_error(p_exact_1, p_pred_1)
residual_1 = np.mean(np.absolute(f_1))
print("Accuracy at t = 0:")
print("Mean residual: ", residual_0)
print("L2 realtive error in u: ", l2_difference_u_0)
print("L2 relative error in v: ", l2_difference_v_0)
print("L2 relative error in w: ", l2_difference_w_0)
print("\n")
print("Accuracy at t = 1:")
print("Mean residual: ", residual_1)
print("L2 realtive error in u: ", l2_difference_u_1)
print("L2 relative error in v: ", l2_difference_v_1)
print("L2 relative error in w: ", l2_difference_w_1)