1准备部分
"""
@author: Maziar Raissi
"""
import sys
sys.path.insert(0, '../../Utilities/')
import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt
import scipy.io
from scipy.interpolate import griddata
from pyDOE import lhs
from plotting import newfig, savefig
from mpl_toolkits.mplot3d import Axes3D
import time
import matplotlib.gridspec as gridspec
from mpl_toolkits.axes_grid1 import make_axes_locatable
np.random.seed(1234)
tf.set_random_seed(1234)
2定义 PhysicsInformedNN:
class PhysicsInformedNN:
# Initialize the class
def __init__(self, X_u, u, X_f, layers, lb, ub, nu):
self.lb = lb
self.ub = ub
self.x_u = X_u[:,0:1]
self.t_u = X_u[:,1:2]
self.x_f = X_f[:,0:1]
self.t_f = X_f[:,1:2]
self.u = u
self.layers = layers
self.nu = nu
# Initialize NNs
self.weights, self.biases = self.initialize_NN(layers)
# tf placeholders and graph
self.sess = tf.Session(config=tf.ConfigProto(allow_soft_placement=True,
log_device_placement=True))
self.x_u_tf = tf.placeholder(tf.float32, shape=[None, self.x_u.shape[1]])
self.t_u_tf = tf.placeholder(tf.float32, shape=[None, self.t_u.shape[1]])
self.u_tf = tf.placeholder(tf.float32, shape=[None, self.u.shape[1]])
self.x_f_tf = tf.placeholder(tf.float32, shape=[None, self.x_f.shape[1]])
self.t_f_tf = tf.placeholder(tf.float32, shape=[None, self.t_f.shape[1]])
self.u_pred = self.net_u(self.x_u_tf, self.t_u_tf)
self.f_pred = self.net_f(self.x_f_tf, self.t_f_tf)
self.loss = tf.reduce_mean(tf.square(self.u_tf - self.u_pred)) + \
tf.reduce_mean(tf.square(self.f_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})
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 net_u(self, x, t):
u = self.neural_net(tf.concat([x,t],1), self.weights, self.biases)
return u
def net_f(self, x,t):
u = self.net_u(x,t)
u_t = tf.gradients(u, t)[0]
u_x = tf.gradients(u, x)[0]
u_xx = tf.gradients(u_x, x)[0]
f = u_t + u*u_x - self.nu*u_xx
return f
def callback(self, loss):
print('Loss:', loss)
def train(self):
tf_dict = {self.x_u_tf: self.x_u, self.t_u_tf: self.t_u, self.u_tf: self.u,
self.x_f_tf: self.x_f, self.t_f_tf: self.t_f}
self.optimizer.minimize(self.sess,
feed_dict = tf_dict,
fetches = [self.loss],
loss_callback = self.callback)
def predict(self, X_star):
u_star = self.sess.run(self.u_pred, {self.x_u_tf: X_star[:,0:1], self.t_u_tf: X_star[:,1:2]})
f_star = self.sess.run(self.f_pred, {self.x_f_tf: X_star[:,0:1], self.t_f_tf: X_star[:,1:2]})
return u_star, f_star
3主程序
3.1 计算
if __name__ == "__main__":
nu = 0.01/np.pi
noise = 0.0
N_u = 100
N_f = 10000
layers = [2, 20, 20, 20, 20, 20, 20, 20, 20, 1]
data = scipy.io.loadmat('../Data/burgers_shock.mat')
t = data['t'].flatten()[:,None]
x = data['x'].flatten()[:,None]
Exact = np.real(data['usol']).T
X, T = np.meshgrid(x,t)
X_star = np.hstack((X.flatten()[:,None], T.flatten()[:,None]))
u_star = Exact.flatten()[:,None]
# Doman bounds
lb = X_star.min(0)
ub = X_star.max(0)
xx1 = np.hstack((X[0:1,:].T, T[0:1,:].T))
uu1 = Exact[0:1,:].T
xx2 = np.hstack((X[:,0:1], T[:,0:1]))
uu2 = Exact[:,0:1]
xx3 = np.hstack((X[:,-1:], T[:,-1:]))
uu3 = Exact[:,-1:]
X_u_train = np.vstack([xx1, xx2, xx3])
X_f_train = lb + (ub-lb)*lhs(2, N_f)
X_f_train = np.vstack((X_f_train, X_u_train))
u_train = np.vstack([uu1, uu2, uu3])
idx = np.random.choice(X_u_train.shape[0], N_u, replace=False)
X_u_train = X_u_train[idx, :]
u_train = u_train[idx,:]
model = PhysicsInformedNN(X_u_train, u_train, X_f_train, layers, lb, ub, nu)
start_time = time.time()
model.train()
elapsed = time.time() - start_time
print('Training time: %.4f' % (elapsed))
u_pred, f_pred = model.predict(X_star)
error_u = np.linalg.norm(u_star-u_pred,2)/np.linalg.norm(u_star,2)
print('Error u: %e' % (error_u))
U_pred = griddata(X_star, u_pred.flatten(), (X, T), method='cubic')
Error = np.abs(Exact - U_pred)
3.2 作图
fig, ax = newfig(1.0, 1.1)
ax.axis('off')
####### Row 0: u(t,x) ##################
gs0 = gridspec.GridSpec(1, 2)
gs0.update(top=1-0.06, bottom=1-1/3, left=0.15, right=0.85, wspace=0)
ax = plt.subplot(gs0[:, :])
h = ax.imshow(U_pred.T, interpolation='nearest', cmap='rainbow',
extent=[t.min(), t.max(), x.min(), x.max()],
origin='lower', aspect='auto')
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(h, cax=cax)
ax.plot(X_u_train[:,1], X_u_train[:,0], 'kx', label = 'Data (%d points)' % (u_train.shape[0]), markersize = 4, clip_on = False)
line = np.linspace(x.min(), x.max(), 2)[:,None]
ax.plot(t[25]*np.ones((2,1)), line, 'w-', linewidth = 1)
ax.plot(t[50]*np.ones((2,1)), line, 'w-', linewidth = 1)
ax.plot(t[75]*np.ones((2,1)), line, 'w-', linewidth = 1)
ax.set_xlabel('$t$')
ax.set_ylabel('$x$')
ax.legend(frameon=False, loc = 'best')
ax.set_title('$u(t,x)$', fontsize = 10)
####### Row 1: u(t,x) slices ##################
gs1 = gridspec.GridSpec(1, 3)
gs1.update(top=1-1/3, bottom=0, left=0.1, right=0.9, wspace=0.5)
ax = plt.subplot(gs1[0, 0])
ax.plot(x,Exact[25,:], 'b-', linewidth = 2, label = 'Exact')
ax.plot(x,U_pred[25,:], 'r--', linewidth = 2, label = 'Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.set_title('$t = 0.25$', fontsize = 10)
ax.axis('square')
ax.set_xlim([-1.1,1.1])
ax.set_ylim([-1.1,1.1])
ax = plt.subplot(gs1[0, 1])
ax.plot(x,Exact[50,:], 'b-', linewidth = 2, label = 'Exact')
ax.plot(x,U_pred[50,:], 'r--', linewidth = 2, label = 'Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.axis('square')
ax.set_xlim([-1.1,1.1])
ax.set_ylim([-1.1,1.1])
ax.set_title('$t = 0.50$', fontsize = 10)
ax.legend(loc='upper center', bbox_to_anchor=(0.5, -0.35), ncol=5, frameon=False)
ax = plt.subplot(gs1[0, 2])
ax.plot(x,Exact[75,:], 'b-', linewidth = 2, label = 'Exact')
ax.plot(x,U_pred[75,:], 'r--', linewidth = 2, label = 'Prediction')
ax.set_xlabel('$x$')
ax.set_ylabel('$u(t,x)$')
ax.axis('square')
ax.set_xlim([-1.1,1.1])
ax.set_ylim([-1.1,1.1])
ax.set_title('$t = 0.75$', fontsize = 10)
# savefig('./figures/Burgers')
源自Maziar
-
Raissi, Maziar, Paris Perdikaris, and George E. Karniadakis. "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations." Journal of Computational Physics 378 (2019): 686-707.
-
Raissi, Maziar, Paris Perdikaris, and George Em Karniadakis. "Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations." arXiv preprint arXiv:1711.10561 (2017).
-
Raissi, Maziar, Paris Perdikaris, and George Em Karniadakis. "Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations." arXiv preprint arXiv:1711.10566 (2017).
Citation
@article{raissi2019physics,
title={Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E},
journal={Journal of Computational Physics},
volume={378},
pages={686--707},
year={2019},
publisher={Elsevier}
}
@article{raissi2017physicsI,
title={Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations},
author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George Em},
journal={arXiv preprint arXiv:1711.10561},
year={2017}
}
@article{raissi2017physicsII,
title={Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations},
author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George Em},
journal={arXiv preprint arXiv:1711.10566},
year={2017}
}
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