-
Notifications
You must be signed in to change notification settings - Fork 1.2k
/
KdV.py
executable file
·313 lines (241 loc) · 12.1 KB
/
KdV.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
"""
@author: Maziar Raissi
"""
import sys
sys.path.insert(0, '../../Utilities/')
import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt
import time
import scipy.io
from plotting import newfig, savefig
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.gridspec as gridspec
from mpl_toolkits.axes_grid1 import make_axes_locatable
np.random.seed(1234)
tf.set_random_seed(1234)
class PhysicsInformedNN:
# Initialize the class
def __init__(self, x0, u0, x1, u1, layers, dt, lb, ub, q):
self.lb = lb
self.ub = ub
self.x0 = x0
self.x1 = x1
self.u0 = u0
self.u1 = u1
self.layers = layers
self.dt = dt
self.q = max(q,1)
# Initialize NN
self.weights, self.biases = self.initialize_NN(layers)
# Initialize parameters
self.lambda_1 = tf.Variable([0.0], dtype=tf.float32)
self.lambda_2 = tf.Variable([-6.0], dtype=tf.float32)
# Load IRK weights
tmp = np.float32(np.loadtxt('../../Utilities/IRK_weights/Butcher_IRK%d.txt' % (q), ndmin = 2))
weights = np.reshape(tmp[0:q**2+q], (q+1,q))
self.IRK_alpha = weights[0:-1,:]
self.IRK_beta = weights[-1:,:]
self.IRK_times = tmp[q**2+q:]
# tf placeholders and graph
self.sess = tf.Session(config=tf.ConfigProto(allow_soft_placement=True,
log_device_placement=True))
self.x0_tf = tf.placeholder(tf.float32, shape=(None, self.x0.shape[1]))
self.x1_tf = tf.placeholder(tf.float32, shape=(None, self.x1.shape[1]))
self.u0_tf = tf.placeholder(tf.float32, shape=(None, self.u0.shape[1]))
self.u1_tf = tf.placeholder(tf.float32, shape=(None, self.u1.shape[1]))
self.dummy_x0_tf = tf.placeholder(tf.float32, shape=(None, self.q)) # dummy variable for fwd_gradients
self.dummy_x1_tf = tf.placeholder(tf.float32, shape=(None, self.q)) # dummy variable for fwd_gradients
self.U0_pred = self.net_U0(self.x0_tf) # N0 x q
self.U1_pred = self.net_U1(self.x1_tf) # N1 x q
self.loss = tf.reduce_sum(tf.square(self.u0_tf - self.U0_pred)) + \
tf.reduce_sum(tf.square(self.u1_tf - self.U1_pred))
self.optimizer = tf.contrib.opt.ScipyOptimizerInterface(self.loss,
method = 'L-BFGS-B',
options = {'maxiter': 50000,
'maxfun': 50000,
'maxcor': 50,
'maxls': 50,
'ftol' : 1.0 * np.finfo(float).eps})
self.optimizer_Adam = tf.train.AdamOptimizer()
self.train_op_Adam = self.optimizer_Adam.minimize(self.loss)
init = tf.global_variables_initializer()
self.sess.run(init)
def initialize_NN(self, layers):
weights = []
biases = []
num_layers = len(layers)
for l in range(0,num_layers-1):
W = self.xavier_init(size=[layers[l], layers[l+1]])
b = tf.Variable(tf.zeros([1,layers[l+1]], dtype=tf.float32), dtype=tf.float32)
weights.append(W)
biases.append(b)
return weights, biases
def xavier_init(self, size):
in_dim = size[0]
out_dim = size[1]
xavier_stddev = np.sqrt(2/(in_dim + out_dim))
return tf.Variable(tf.truncated_normal([in_dim, out_dim], stddev=xavier_stddev), dtype=tf.float32)
def neural_net(self, X, weights, biases):
num_layers = len(weights) + 1
H = 2.0*(X - self.lb)/(self.ub - self.lb) - 1.0
for l in range(0,num_layers-2):
W = weights[l]
b = biases[l]
H = tf.tanh(tf.add(tf.matmul(H, W), b))
W = weights[-1]
b = biases[-1]
Y = tf.add(tf.matmul(H, W), b)
return Y
def fwd_gradients_0(self, U, x):
g = tf.gradients(U, x, grad_ys=self.dummy_x0_tf)[0]
return tf.gradients(g, self.dummy_x0_tf)[0]
def fwd_gradients_1(self, U, x):
g = tf.gradients(U, x, grad_ys=self.dummy_x1_tf)[0]
return tf.gradients(g, self.dummy_x1_tf)[0]
def net_U0(self, x):
lambda_1 = self.lambda_1
lambda_2 = tf.exp(self.lambda_2)
U = self.neural_net(x, self.weights, self.biases)
U_x = self.fwd_gradients_0(U, x)
U_xx = self.fwd_gradients_0(U_x, x)
U_xxx = self.fwd_gradients_0(U_xx, x)
F = -lambda_1*U*U_x - lambda_2*U_xxx
U0 = U - self.dt*tf.matmul(F, self.IRK_alpha.T)
return U0
def net_U1(self, x):
lambda_1 = self.lambda_1
lambda_2 = tf.exp(self.lambda_2)
U = self.neural_net(x, self.weights, self.biases)
U_x = self.fwd_gradients_1(U, x)
U_xx = self.fwd_gradients_1(U_x, x)
U_xxx = self.fwd_gradients_1(U_xx, x)
F = -lambda_1*U*U_x - lambda_2*U_xxx
U1 = U + self.dt*tf.matmul(F, (self.IRK_beta - self.IRK_alpha).T)
return U1
def callback(self, loss):
print('Loss:', loss)
def train(self, nIter):
tf_dict = {self.x0_tf: self.x0, self.u0_tf: self.u0,
self.x1_tf: self.x1, self.u1_tf: self.u1,
self.dummy_x0_tf: np.ones((self.x0.shape[0], self.q)),
self.dummy_x1_tf: np.ones((self.x1.shape[0], self.q))}
start_time = time.time()
for it in range(nIter):
self.sess.run(self.train_op_Adam, tf_dict)
# Print
if it % 10 == 0:
elapsed = time.time() - start_time
loss_value = self.sess.run(self.loss, tf_dict)
lambda_1_value = self.sess.run(self.lambda_1)
lambda_2_value = np.exp(self.sess.run(self.lambda_2))
print('It: %d, Loss: %.3e, l1: %.3f, l2: %.5f, Time: %.2f' %
(it, loss_value, lambda_1_value, lambda_2_value, elapsed))
start_time = time.time()
self.optimizer.minimize(self.sess,
feed_dict = tf_dict,
fetches = [self.loss],
loss_callback = self.callback)
def predict(self, x_star):
U0_star = self.sess.run(self.U0_pred, {self.x0_tf: x_star, self.dummy_x0_tf: np.ones((x_star.shape[0], self.q))})
U1_star = self.sess.run(self.U1_pred, {self.x1_tf: x_star, self.dummy_x1_tf: np.ones((x_star.shape[0], self.q))})
return U0_star, U1_star
if __name__ == "__main__":
q = 50
skip = 120
N0 = 199
N1 = 201
layers = [1, 50, 50, 50, 50, q]
data = scipy.io.loadmat('../Data/KdV.mat')
t_star = data['tt'].flatten()[:,None]
x_star = data['x'].flatten()[:,None]
Exact = np.real(data['uu'])
idx_t = 40
######################################################################
######################## Noiseles Data ###############################
######################################################################
noise = 0.0
idx_x = np.random.choice(Exact.shape[0], N0, replace=False)
x0 = x_star[idx_x,:]
u0 = Exact[idx_x,idx_t][:,None]
u0 = u0 + noise*np.std(u0)*np.random.randn(u0.shape[0], u0.shape[1])
idx_x = np.random.choice(Exact.shape[0], N1, replace=False)
x1 = x_star[idx_x,:]
u1 = Exact[idx_x,idx_t + skip][:,None]
u1 = u1 + noise*np.std(u1)*np.random.randn(u1.shape[0], u1.shape[1])
dt = np.asscalar(t_star[idx_t+skip] - t_star[idx_t])
# Doman bounds
lb = x_star.min(0)
ub = x_star.max(0)
model = PhysicsInformedNN(x0, u0, x1, u1, layers, dt, lb, ub, q)
model.train(nIter = 50000)
U0_pred, U1_pred = model.predict(x_star)
lambda_1_value = model.sess.run(model.lambda_1)
lambda_2_value = np.exp(model.sess.run(model.lambda_2))
error_lambda_1 = np.abs(lambda_1_value - 1.0)/1.0 *100
error_lambda_2 = np.abs(lambda_2_value - 0.0025)/0.0025 * 100
print('Error lambda_1: %f%%' % (error_lambda_1))
print('Error lambda_2: %f%%' % (error_lambda_2))
######################################################################
########################### Noisy Data ###############################
######################################################################
noise = 0.01
u0 = u0 + noise*np.std(u0)*np.random.randn(u0.shape[0], u0.shape[1])
u1 = u1 + noise*np.std(u1)*np.random.randn(u1.shape[0], u1.shape[1])
model = PhysicsInformedNN(x0, u0, x1, u1, layers, dt, lb, ub, q)
model.train(nIter = 50000)
U_pred = model.predict(x_star)
U0_pred, U1_pred = model.predict(x_star)
lambda_1_value_noisy = model.sess.run(model.lambda_1)
lambda_2_value_noisy = np.exp(model.sess.run(model.lambda_2))
error_lambda_1_noisy = np.abs(lambda_1_value_noisy - 1.0)/1.0 *100
error_lambda_2_noisy = np.abs(lambda_2_value_noisy - 0.0025)/0.0025 * 100
print('Error lambda_1: %f%%' % (error_lambda_1_noisy))
print('Error lambda_2: %f%%' % (error_lambda_2_noisy))
######################################################################
############################# Plotting ###############################
######################################################################
fig, ax = newfig(1.0, 1.5)
ax.axis('off')
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1-0.06, bottom=1-1/3+0.05, left=0.15, right=0.85, wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(Exact, interpolation='nearest', cmap='rainbow',
extent=[t_star.min(),t_star.max(), lb[0], ub[0]],
origin='lower', aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
line = np.linspace(x_star.min(), x_star.max(), 2)[:,None]
ax.plot(t_star[idx_t]*np.ones((2,1)), line, 'w-', linewidth = 1.0)
ax.plot(t_star[idx_t + skip]*np.ones((2,1)), line, 'w-', linewidth = 1.0)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
ax.set_title('$u(t,x)$', fontsize = 10)
gs1 = gridspec.GridSpec(1, 2)
gs1.update(top=1-1/3-0.1, bottom=1-2/3, left=0.15, right=0.85, wspace=0.5)
ax = plt.subplot(gs1[0, 0])
ax.plot(x_star,Exact[:,idx_t][:,None], 'b', linewidth = 2, label = 'Exact')
ax.plot(x0, u0, 'rx', linewidth = 2, label = 'Data')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = %.2f$\n%d trainng data' % (t_star[idx_t], u0.shape[0]), fontsize = 10)
ax = plt.subplot(gs1[0, 1])
ax.plot(x_star,Exact[:,idx_t + skip][:,None], 'b', linewidth = 2, label = 'Exact')
ax.plot(x1, u1, 'rx', linewidth = 2, label = 'Data')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = %.2f$\n%d trainng data' % (t_star[idx_t+skip], u1.shape[0]), fontsize = 10)
ax.legend(loc='upper center', bbox_to_anchor=(-0.3, -0.3), ncol=2, frameon=False)
gs2 = gridspec.GridSpec(1, 2)
gs2.update(top=1-2/3-0.05, bottom=0, left=0.15, right=0.85, wspace=0.0)
ax = plt.subplot(gs2[0, 0])
ax.axis('off')
s1 = r'$\begin{tabular}{ |c|c| } \hline Correct PDE & $u_t + u u_x + 0.0025 u_{xxx} = 0$ \\ \hline Identified PDE (clean data) & '
s2 = r'$u_t + %.3f u u_x + %.7f u_{xxx} = 0$ \\ \hline ' % (lambda_1_value, lambda_2_value)
s3 = r'Identified PDE (1\% noise) & '
s4 = r'$u_t + %.3f u u_x + %.7f u_{xxx} = 0$ \\ \hline ' % (lambda_1_value_noisy, lambda_2_value_noisy)
s5 = r'\end{tabular}$'
s = s1+s2+s3+s4+s5
ax.text(-0.1,0.2,s)
# savefig('./figures/KdV')