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 feature_transform(t):
t = 0.01 * t
return tf.concat(
(t, tf.sin(t), tf.sin(2 * t), tf.sin(3 * t), tf.sin(
4 * t), tf.sin(5 * t)),
axis=1,
)
def feature_transform(inputs):
# Periodic BC in x
P = BOX[1][0] - BOX[0][0] + 2 * DPML
w = 2 * np.pi / P
x, y = w * inputs[:, :1], inputs[:, 1:]
return tf.concat(
(
tf.math.cos(x),
tf.math.sin(x),
tf.math.cos(2 * x),
tf.math.sin(2 * x),
tf.math.cos(3 * x),
tf.math.sin(3 * x),
tf.math.cos(4 * x),
tf.math.sin(4 * x),
tf.math.cos(5 * x),
tf.math.sin(5 * x),
tf.math.cos(6 * x),
tf.math.sin(6 * x),
# tf.math.cos(7 * x),
# tf.math.sin(7 * x),
# tf.math.cos(8 * x),
# tf.math.sin(8 * x),
# tf.math.cos(9 * x),
# tf.math.sin(9 * x),
# tf.math.cos(10 * x),
# tf.math.sin(10 * x),
y,
tf.math.cos(OMEGA * y),
tf.math.sin(OMEGA * y),
),
axis=1,
)
def modify_inv_heter(self, x, y):
domain_space = x[:, 0:2]
D = tf.layers.dense(tf.layers.dense(
tf.layers.dense(
tf.layers.dense(
tf.layers.dense(
tf.layers.dense(domain_space, 60, tf.nn.tanh), 60,
tf.nn.tanh), 60, tf.nn.tanh), 60, tf.nn.tanh), 60,
tf.nn.tanh),
1,
activation=None)
return tf.concat((y[:, 0:2], D), axis=1)
def feature_transform(t):
return tf.concat(
(
t,
tf.sin(t),
tf.sin(2 * t),
tf.sin(3 * t),
tf.sin(4 * t),
tf.sin(5 * t),
tf.sin(6 * t),
),
axis=1,
)
def initialState_tf(X, wei=-1, sample_output=None):
"""
initial state for the rho 2020/12/25 wodnering
ρ(q,0)=1/[〖(2π)〗^(1/4) √σ]^N |├ UP⟩⟨UP┤|∙∏_(s=1)^N▒〖ψ_((α_s=0)) (q_s)〗
ρ(q,0)=1/[〖(2π)〗^(1/4) √σ]^N ∑_(i=1)^(N+1)▒∑_(j=1)^(N+1)▒〖ρ_ij (0)〗|├ i⟩⟨j┤|∙∏_(s=1)^N▒〖ψ_((α_s=0)) (q_s)〗
ρ_ij (0)=1/2N (1-δ_(i,N+1) )(1-δ_(j,N+1) )+1/2 δ_(i,N+1) δ_(j,N+1)
+1/(2√N) [(1-δ_(i,N+1) ) δ_(j,N+1)+δ_(i,N+1) (1-δ_(j,N+1) )]
ψ_((α_s=0) ) (q_s )=e^(-q_s^2/2σ^2 )/(〖(2π)〗^(1/4) √σ) H_0 (q_s/(√2 σ))
"""
def rhoij(i, j):
"""
get the rho(i,j) for inital state
"""
assert (i >= j)
if i == j:
if i == N + 1:
Rhoij = 1 / 2
else:
Rhoij = 1 / 2 / Ntf
elif i == N + 1:
Rhoij = 1 / 2 / (Ntf)**0.5
else:
Rhoij = 1 / 2 / Ntf
return Rhoij
if sample_output is None:
NUM_initial, _ = np.shape(X)
Rho0 = np.zeros([NUM_initial, C.sizeRho])
else:
Rho0 = []
for i in range(0, C.sizeRho):
Rho0.append(sample_output[:, i:i + 1] * 0)
N = C.N
ConstantNumber = 1 / ((2 * pi)**(1 / 4) * (sigma)**0.5)**Ntf
Phi0_multi = 1
for s in range(1, N):
Phi0_multi = Phi0_multi * phi(X[:, s:s + 1], 0)
for j in range(1, C.N + 2):
for i in range(j + 1, C.N + 2):
Xnum = getXHalf_real(i, j, C.N)
X_imag = getXHalf_imag(i, j, C.N)
Rho0[Xnum] = Rho0[Xnum] + ConstantNumber * Phi0_multi * rhoij(i, j)
for n in range(1, N + 2):
Rho0[n - 1] = Rho0[n - 1] + ConstantNumber * Phi0_multi * rhoij(n, n)
if wei == -1:
return tf.concat(Rho0, axis=1)
else:
return Rho0[:, wei:wei + 1]
def modify_heter(self, x, y):
x_space, y_space = x[:, 0:1], x[:, 1:2]
x_upper = tf.less_equal(x_space, 54 * 0.1)
x_lower = tf.greater(x_space, 32 * 0.1)
cond_1 = tf.logical_and(x_upper, x_lower)
y_upper = tf.less_equal(y_space, 54 * 0.1)
y_lower = tf.greater(y_space, 32 * 0.1)
cond_2 = tf.logical_and(y_upper, y_lower)
D0 = tf.ones_like(x_space) * 0.02
D1 = tf.ones_like(x_space) * 0.1
D = tf.where(tf.logical_and(cond_1, cond_2), D0, D1)
return tf.concat((y[:, 0:2], D), axis=1)
def output_transform(inputs, outputs):
x, y = inputs[:, :1], inputs[:, 1:]
# 1 <= eps <= 12
eps = 1 + 11 * tf.math.sigmoid(outputs[:, -1:])
# Zero Dirichlet BC
a, b = BOX[0][1] - DPML, BOX[1][1] + DPML
E = (1 - tf.math.exp(a - y)) * (1 - tf.math.exp(y - b)) * outputs[:, :2]
# Zero Dirichlet and Neumann BC
# a, b = BOX[0][1] - DPML, BOX[1][1] + DPML
# E = 0.01 * (a - y) ** 2 * (y - b) ** 2 * outputs[:, :2]
# return E
return tf.concat((E, eps), axis=1)
def initial_condition(x, y):
"""
main function for differential function with q
"""
def rhoij(i, j):
"""
get the rho(i,j) for inital state
"""
assert (i >= j)
if i == j:
if i == C.N + 1:
Rhoij = 1 / 2
else:
Rhoij = 1 / 2 / Ntf
elif i == C.N + 1:
Rhoij = 1 / 2 / (Ntf)**0.5
else:
Rhoij = 1 / 2 / Ntf
return Rhoij
Rho0 = []
for i in range(0, C.sizeRho):
Rho0.append(y[:, i:i + 1] * 0)
Phi0_multi = etf**(-x[:, 0:1]**2 /
(2 * sigma**2)) / ((2 * pi)**0.5 * sigma)**0.5
for s in range(1, C.N):
Phi0_multi = Phi0_multi * etf**(-x[:, s:s + 1]**2 / (2 * sigma**2)) / (
(2 * pi)**0.5 * sigma)**0.5
for n in range(1, C.N + 2):
Rho0[n - 1] = Rho0[n - 1] + ConstantNumber * Phi0_multi * rhoij(n, n)
for j in range(1, C.N + 1):
for i in range(j + 1, C.N + 1):
Xnum = getXHalf_real(i, j, C.N)
Rho0[Xnum] = Rho0[Xnum] * ConstantNumber * Phi0_multi * rhoij(i, j)
result = []
for i in range(0, C.sizeRho):
result.append(x[:, C.N:] * y[:, i:i + 1] + Rho0[i])
for j in range(0, C.N):
result[i] = result[i] * (x[:, j:j + 1] - QmaxTF) * (x[:, j:j + 1] +
QmaxTF)
return tf.concat(result, axis=1)
def output_transform(inputs, outputs):
x, y = inputs[:, :1], inputs[:, 1:]
bc = 16 * x * (1 - x) * y * (1 - y)
# u
u0 = 1
u = tf.math.abs(u0 + bc * outputs[:, :1])
# v
v = bc * outputs[:, 1:2]
# p
p = (1 - x) * outputs[:, 2:3]
# rho
# rho = tf.math.exp(-bc * tf.math.square(outputs[:, 3:]))
# rho = 1 + bc * outputs[:, 3:]
center = tf.math.square(x - 0.5) + tf.math.square(y - 0.5)
# rho = center * outputs[:, 3:]
rho = center * (bc * outputs[:, 3:] + (1 - bc) * (1 + 1e-6 / 0.25) /
(center + 1e-6))
rho = tf.math.maximum(0.0, tf.math.minimum(1.0, rho))
return tf.concat((u, v, p, rho), axis=1)
def feature_transform(t):
t = 0.1 * t
return tf.concat((t, tf.exp(-t)), axis=1,)
def SLE_q(x, y, C):
"""
SLE function with q
"""
Drho = []
for allindex in range(0, C.sizeRho):
DrhoQ = Gamma / 2 * y[:, allindex:allindex + 1] * Ntf
for qs in range(0, C.N):
DrhoQ = DrhoQ + Gamma / 2 * (
x[:, qs:qs + 1] * dde.grad.jacobian(y, x, i=allindex, j=qs) +
(0.5 + n_bar) *
dde.grad.hessian(y, x, i=qs, j=qs, component=allindex))
DrhoT = dde.grad.jacobian(y, x, i=allindex, j=C.N)
Drho.append(DrhoQ - DrhoT)
# Drho.append(DrhoT)
# for n in range(1 ,C.N + 1):
# # n = 1: N
# for m in range(n + 1 , C.N + 2 ):
# # m = n + 1: N + 1
# X_imag = getXHalf_imag(m,n, C.N)
# X_real = getXHalf_real(m,n, C.N)
# # real part
# indexAA = getXHalf_imag(C.N + 1, m , C.N)
# AA = - (
# y[:,X_imag:X_imag+1] * ( w - v -lamb * w_vib * x[:,n-1:n] )
# - G * y[:,indexAA:indexAA+1]
# )
# DD = AA *x[:,n-1:n] *0
# if m == C.N+1:
# for k in range(1,C.N+1):
# if k >n:
# indexDD = getXHalf_imag(k,n,C.N)
# DD = DD + G * y[:,indexDD:indexDD+1]
# elif k < n:
# indexDD = getXHalf_imag(n,k,C.N)
# DD = DD - G * y[:,indexDD:indexDD+1]
# else:
# indexDD = getXHalf_imag(C.N+1, n ,C.N)
# DD = y[:,X_imag:X_imag+1] * \
# (w - v - lamb * w_vib * x[:,m-1:m] ) \
# + G * y[:, indexDD:indexDD+1]
# # imag part
# indexEE = getXHalf_real(C.N+1,m,C.N )
# EE = y[:,X_real:X_real+1] * (w - v - lamb * w_vib * x[:, n-1:n]) \
# + G * y[:,indexEE:indexEE+1 ]
# HH = EE * 0
# if m == C.N +1 :
# for k in range(1,C.N + 1):
# if k > n:
# indexEE = getXHalf_real(k,n,C.N)
# HH = HH - G * y[:,indexEE :indexEE +1]
# elif k < n:
# indexEE = getXHalf_real(n,k,C.N)
# HH = HH - G * y[:,indexEE:indexEE+1]
# elif k == n:
# HH = HH - G * y[:,n-1:n]
# else:
# indexEE = getXHalf_real(C.N + 1,n,C.N)
# HH = - y[:, X_real:X_real+1]* \
# (w - v - lamb * w_vib * x[:, m-1:m]) \
# - G * y[:,indexEE:indexEE+1]
# Drho[X_real] = Drho[X_real] + AA + DD
# Drho[X_imag] = Drho[X_imag] + EE + HH
return tf.concat(Drho, axis=1)
def feature_transform(x):
return tf.concat(
[x[:, 0:1] * tf.sin(x[:, 1:2]), x[:, 0:1] * tf.cos(x[:, 1:2])], axis=1)