raissi2019 Navier–Stokes equation（二维） （待更新...）

首先了解一下Navier–Stokes equation
如何通俗形象的解释纳维-斯托克斯方程？ - Porcupine的回答 - 知乎
https://www.zhihu.com/question/332963536/answer/740232439

不考虑二维，只单单看算法结构，发现与\(\lambda\)已知的情况不一样的地方如下。

        # Initialize parameters
        self.lambda_1 = tf.Variable([0.0], dtype=tf.float32)
        self.lambda_2 = tf.Variable([0.0], dtype=tf.float32)

    def net_NS(self, x, y, t):
        lambda_1 = self.lambda_1
        lambda_2 = self.lambda_2
        
        psi_and_p = self.neural_net(tf.concat([x,y,t], 1), self.weights, self.biases)
        psi = psi_and_p[:,0:1]
        p = psi_and_p[:,1:2]
        
        u = tf.gradients(psi, y)[0]
        v = -tf.gradients(psi, x)[0]  
        
        u_t = tf.gradients(u, t)[0]
        u_x = tf.gradients(u, x)[0]
        u_y = tf.gradients(u, y)[0]
        u_xx = tf.gradients(u_x, x)[0]
        u_yy = tf.gradients(u_y, y)[0]
        
        v_t = tf.gradients(v, t)[0]
        v_x = tf.gradients(v, x)[0]
        v_y = tf.gradients(v, y)[0]
        v_xx = tf.gradients(v_x, x)[0]
        v_yy = tf.gradients(v_y, y)[0]
        
        p_x = tf.gradients(p, x)[0]
        p_y = tf.gradients(p, y)[0]

        f_u = u_t + lambda_1*(u*u_x + v*u_y) + p_x - lambda_2*(u_xx + u_yy) 
        f_v = v_t + lambda_1*(u*v_x + v*v_y) + p_y - lambda_2*(v_xx + v_yy)
        
        return u, v, p, f_u, f_v
    

    def train(self, nIter): 

        tf_dict = {self.x_tf: self.x, self.y_tf: self.y, self.t_tf: self.t,
                   self.u_tf: self.u, self.v_tf: self.v}
        
        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 = 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()
            
        # 如何让神经网络知道lambda_1和lambda_2是需要调整的参数呢？我猜想是因为这里的fetches. 猜想不一定对，如有错误，还望不吝指出。
        self.optimizer.minimize(self.sess,
                                feed_dict = tf_dict,
                                fetches = [self.loss, self.lambda_1, self.lambda_2], 
                                loss_callback = self.callback)