code: """
  @author: Maziar Raissi
  """
  
  import sys
  sys.path.insert(0, '../../Utilities/')
  
  import tensorflow as tf
  import numpy as np
  import time
  import scipy.io
  
  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)
          F = -lambda_1*U*U_x + lambda_2*U_xx
          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)
          F = -lambda_1*U*U_x + lambda_2*U_xx
          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
  
      
  def main_loop(skip, noise, num_layers, num_neurons):
          
      N0 = 199
      N1 = 201
          
      data = scipy.io.loadmat('../Data/burgers_shock.mat')
      
      t_star = data['t'].flatten()[:,None]
      x_star = data['x'].flatten()[:,None]
      Exact = np.real(data['usol'])
      
      idx_t = 10
          
      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])        
      q = int(np.ceil(0.5*np.log(np.finfo(float).eps)/np.log(dt)))
      
      layers = np.concatenate([[1], num_neurons*np.ones(num_layers), [q]]).astype(int).tolist()    
      
      # 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))
                  
      nu = 0.01/np.pi       
      error_lambda_1 = np.abs(lambda_1_value - 1.0)/1.0 *100
      error_lambda_2 = np.abs(lambda_2_value - nu)/nu * 100
      
      print('Error lambda_1: %f%%' % (error_lambda_1))
      print('Error lambda_2: %f%%' % (error_lambda_2))
      
      return error_lambda_1, error_lambda_2
      
      
  if __name__ == "__main__": 
      
      skip = [20, 40, 60, 80]
      noise = [0.0, 0.01, 0.05, 0.1]
      
      num_layers = [1,2,3,4]
      num_neurons = [10,25,50]
      
      error_lambda_1_table_1 = np.zeros((len(skip), len(noise)))
      error_lambda_2_table_1 = np.zeros((len(skip), len(noise)))
      
      error_lambda_1_table_2 = np.zeros((len(num_layers), len(num_neurons)))
      error_lambda_2_table_2 = np.zeros((len(num_layers), len(num_neurons)))
      
      for i in range(len(skip)):
          for j in range(len(noise)):
              error_lambda_1_table_1[i,j], error_lambda_2_table_1[i,j] = main_loop(skip[i], noise[j], num_layers[-1], num_neurons[-1])
               
      for i in range(len(num_layers)):
          for j in range(len(num_neurons)):
              error_lambda_1_table_2[i,j], error_lambda_2_table_2[i,j] = main_loop(skip[-1], noise[0], num_layers[i], num_neurons[j])
              
              
      np.savetxt('./tables/error_lambda_1_table_1.csv', error_lambda_1_table_1, delimiter=' & ', fmt='$%2.3f$', newline=' \\\\\n')
      np.savetxt('./tables/error_lambda_2_table_1.csv', error_lambda_2_table_1, delimiter=' & ', fmt='$%2.3f$', newline=' \\\\\n')
  
      np.savetxt('./tables/error_lambda_1_table_2.csv', error_lambda_1_table_2, delimiter=' & ', fmt='$%2.3f$', newline=' \\\\\n')
      np.savetxt('./tables/error_lambda_2_table_2.csv', error_lambda_2_table_2, delimiter=' & ', fmt='$%2.3f$', newline=' \\\\\n')
  
  
apis: numpy.ones numpy.savetxt numpy.random.seed tensorflow.matmul numpy.log tensorflow.Variable tensorflow.global_variables_initializer numpy.asscalar numpy.reshape numpy.random.choice numpy.abs tensorflow.gradients tensorflow.set_random_seed numpy.std numpy.finfo tensorflow.ConfigProto tensorflow.placeholder tensorflow.zeros tensorflow.truncated_normal tensorflow.train.AdamOptimizer numpy.random.randn tensorflow.exp tensorflow.square numpy.sqrt numpy.real numpy.loadtxt