PINN解偏微分方程实例2之一维非线性薛定谔方程

• 1. 一维非线性薛定谔方程
• 2. 损失函数如下定义
• 3. 代码
• 4. 实验细节及复现结果
• 参考资料

1. 一维非线性薛定谔方程

   考虑偏微分方程如下：
iht+0.5hxx+∣h∣2h=0h(0,x)=2sech(x)h(t,−5)=h(t,5)hx(t,−5)=hx(t,5)\begin{align} \begin{aligned} & ih_t + 0.5h_{xx} + |h|^2h = 0 \\ & h(0,x) = 2 sech(x) \\ & h(t,-5) = h(t,5) \\ & h_x(t,-5) = h_x(t,5) \end{aligned} \end{align} ​iht​+0.5hxx​+∣h∣2h=0h(0,x)=2sech(x)h(t,−5)=h(t,5)hx​(t,−5)=hx​(t,5)​​​
其中x∈[−5,5],t∈[0,π/2].x\in[-5,5],t\in[0,\pi/2].x∈[−5,5],t∈[0,π/2].这是一个带有周期性边界条件，初始条件和复数解的偏微分方程。

2. 损失函数如下定义

MSE=MSE0+MSEb+MSEf\begin{align} \begin{aligned} MSE = MSE_0 + MSE_b + MSE_f \\ \end{aligned} \end{align} MSE=MSE0​+MSEb​+MSEf​​​​
其中
MSE0=1N0∑i=1N0∣h(0,x0i)−h0i∣2MSEb=1Nb∑i=1Nb(∣hi(tbi,−5)−hi(tbi,5)∣2+∣hxi(tbi,−5)−hxi(tbi,5)∣2)MSEf=1Nf∑i=1Nf∣f(tfi,xfi)∣2\begin{align} \begin{aligned} MSE_0 &= \frac{1}{N_0}\sum_{i=1}^{N_0}|h(0,x_0^i)-h_0^i|^2 \\ MSE_b &= \frac{1}{N_b}\sum_{i=1}^{N_b}(|h^i(t_b^i,-5)-h^i(t_b^i,5)|^2+|h^i_x(t_b^i,-5)-h^i_x(t_b^i,5)|^2) \\ MSE_f &= \frac{1}{N_f}\sum_{i=1}^{N_f}|f(t_f^i,x_f^i)|^2 \\ \end{aligned} \end{align} MSE0​MSEb​MSEf​​=N0​1​i=1∑N0​​∣h(0,x0i​)−h0i​∣2=Nb​1​i=1∑Nb​​(∣hi(tbi​,−5)−hi(tbi​,5)∣2+∣hxi​(tbi​,−5)−hxi​(tbi​,5)∣2)=Nf​1​i=1∑Nf​​∣f(tfi​,xfi​)∣2​​​
这里MSE0MSE_0MSE0​是初始条件损失函数，MSEbMSE_bMSEb​是周期条件损失函数，MSEfMSE_fMSEf​是偏微分方程构造的损失函数。
  由于h(t,x)=u(t,x)+iv(t,x)h(t,x)=u(t,x)+iv(t,x)h(t,x)=u(t,x)+iv(t,x)，在代码实现过程中，损失函数具体形式如下：
l=l1+l2+l3+l4+l5+l6+l7+l8\begin{align} \begin{aligned} l = l_1 + l_2 + l_3 + l_4 + l_5 + l_6 + l_7 + l_8 \end{aligned} \end{align} l=l1​+l2​+l3​+l4​+l5​+l6​+l7​+l8​​​​
其中
l1=1N0∑i=1N0∣u(0,x0i)−u0i∣2l2=1N0∑i=1N0∣v(0,x0i)−v0i∣2l3=1Nb∑i=1Nb∣ui(tbi,−5)−ui(tbi,5)∣2l4=1Nb∑i=1Nb∣vi(tbi,−5)−vi(tbi,5)∣2l5=1Nb∑i=1Nb∣uxi(tbi,−5)−uxi(tbi,5)∣2l6=1Nb∑i=1Nb∣vxi(tbi,−5)−vxi(tbi,5)∣2l7=1Nf∑i=1Nf∣ut+0.5∗vxx+(u2+v2)∗v∣2l8=1Nf∑i=1Nf∣vt+0.5∗uxx+(u2+v2)∗u∣2\begin{align} \begin{aligned} l_1 &= \frac{1}{N_0}\sum_{i=1}^{N_0}|u(0,x_0^i)-u_0^i|^2 \\ l_2 &= \frac{1}{N_0}\sum_{i=1}^{N_0}|v(0,x_0^i)-v_0^i|^2 \\ l_3 &= \frac{1}{N_b}\sum_{i=1}^{N_b}|u^i(t_b^i,-5)-u^i(t_b^i,5)|^2 \\ l_4 &= \frac{1}{N_b}\sum_{i=1}^{N_b}|v^i(t_b^i,-5)-v^i(t_b^i,5)|^2 \\ l_5 &= \frac{1}{N_b}\sum_{i=1}^{N_b}|u^i_x(t_b^i,-5)-u^i_x(t_b^i,5)|^2 \\ l_6 &= \frac{1}{N_b}\sum_{i=1}^{N_b}|v^i_x(t_b^i,-5)-v^i_x(t_b^i,5)|^2 \\ l_7 &= \frac{1}{N_f}\sum_{i=1}^{N_f}|u_t + 0.5 *v _{xx} + (u^2+v^2)*v|^2 \\ l_8 &= \frac{1}{N_f}\sum_{i=1}^{N_f}|v_t + 0.5 *u _{xx} + (u^2+v^2)*u|^2 \\ \end{aligned} \end{align} l1​l2​l3​l4​l5​l6​l7​l8​​=N0​1​i=1∑N0​​∣u(0,x0i​)−u0i​∣2=N0​1​i=1∑N0​​∣v(0,x0i​)−v0i​∣2=Nb​1​i=1∑Nb​​∣ui(tbi​,−5)−ui(tbi​,5)∣2=Nb​1​i=1∑Nb​​∣vi(tbi​,−5)−vi(tbi​,5)∣2=Nb​1​i=1∑Nb​​∣uxi​(tbi​,−5)−uxi​(tbi​,5)∣2=Nb​1​i=1∑Nb​​∣vxi​(tbi​,−5)−vxi​(tbi​,5)∣2=Nf​1​i=1∑Nf​​∣ut​+0.5∗vxx​+(u2+v2)∗v∣2=Nf​1​i=1∑Nf​​∣vt​+0.5∗uxx​+(u2+v2)∗u∣2​​​
这里N0=Nb=50,Nf=20000.N_0=N_b=50,N_f=20000.N0​=Nb​=50,Nf​=20000. 其中u0i,v0iu_0^i,v_0^iu0i​,v0i​为谱方法计算出来的真解，其它均为神经网络输出值。

3. 代码

  代码参考https://github.com/maziarraissi/PINNs,原代码运行框架tensorflow1，这里将其改为tensorflow2上运行，代码如下：

"""
@author: Maziar Raissi
@Annotator：suntao
利用谱方法计算了t*x为[0,pi/2]*[-5,5]区域上的真解，真解个数t*x为201*256
"""import sys
sys.path.insert(0, '../../Utilities/')import tensorflow.compat.v1 as tf   # tensorflow1.0代码迁移到2.0上运行，加上这两行
tf.disable_v2_behavior()import tensorflow as tf2
import numpy as np
import matplotlib.pyplot as plt
import scipy.io
from scipy.interpolate import griddata
from pyDOE import lhs    # 拉丁超立方采样
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_locatablenp.random.seed(1234)
tf.random.set_random_seed(1234)    # tf.random.set_seedclass PhysicsInformedNN:# Initialize the classdef __init__(self, x0, u0, v0, tb, X_f, layers, lb, ub):"""由损失函数可以看出，周期性边界不需要真解，仅初始条件需要真解，就可以求解:param x0: 左边界N0个点x值:param u0: 左边界N0个点对应解的实部:param v0: 左边界N0个点对应解的虚部:param tb: 周期边界Nb个点t值:param X_f: 在区域[0,pi/2]*[-5,5]内采用拉丁超立方采样得到的Nf个点的坐标值:param layers: 神经网络各层神经元列表:param lb: np.array([-5.0, 0.0]):param ub: np.array([5.0, np.pi/2])"""X0 = np.concatenate((x0, 0*x0), 1)  # 左边界坐标点(x,0)X_lb = np.concatenate((0*tb + lb[0], tb), 1)  # 下边界坐标点(-5,t)X_ub = np.concatenate((0*tb + ub[0], tb), 1)  # 上边界坐标点(5,t)self.lb = lbself.ub = ubself.x0 = X0[:,0:1]self.t0 = X0[:,1:2]self.x_lb = X_lb[:,0:1]self.t_lb = X_lb[:,1:2]self.x_ub = X_ub[:,0:1]self.t_ub = X_ub[:,1:2]self.x_f = X_f[:,0:1]self.t_f = X_f[:,1:2]self.u0 = u0self.v0 = v0# Initialize NNsself.layers = layersself.weights, self.biases = self.initialize_NN(layers)# tf Placeholdersself.x0_tf = tf.placeholder(tf.float32, shape=[None, self.x0.shape[1]])    # tf.placeholder改为tf.compat.v1.placeholderself.t0_tf = tf.placeholder(tf.float32, shape=[None, self.t0.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.u0_tf = tf.placeholder(tf.float32, shape=[None, self.u0.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.v0_tf = tf.placeholder(tf.float32, shape=[None, self.v0.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.x_lb_tf = tf.placeholder(tf.float32, shape=[None, self.x_lb.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.t_lb_tf = tf.placeholder(tf.float32, shape=[None, self.t_lb.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.x_ub_tf = tf.placeholder(tf.float32, shape=[None, self.x_ub.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.t_ub_tf = tf.placeholder(tf.float32, shape=[None, self.t_ub.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.x_f_tf = tf.placeholder(tf.float32, shape=[None, self.x_f.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholderself.t_f_tf = tf.placeholder(tf.float32, shape=[None, self.t_f.shape[1]])     # tf.placeholder改为tf.compat.v1.placeholder# tf Graphsself.u0_pred, self.v0_pred, _ , _ = self.net_uv(self.x0_tf, self.t0_tf)    # 左边界self.u_lb_pred, self.v_lb_pred, self.u_x_lb_pred, self.v_x_lb_pred = self.net_uv(self.x_lb_tf, self.t_lb_tf)    # 下边界self.u_ub_pred, self.v_ub_pred, self.u_x_ub_pred, self.v_x_ub_pred = self.net_uv(self.x_ub_tf, self.t_ub_tf)    # 上边界self.f_u_pred, self.f_v_pred = self.net_f_uv(self.x_f_tf, self.t_f_tf)# Lossself.loss = tf.reduce_mean(tf.square(self.u0_tf - self.u0_pred)) + \tf.reduce_mean(tf.square(self.v0_tf - self.v0_pred)) + \tf.reduce_mean(tf.square(self.u_lb_pred - self.u_ub_pred)) + \tf.reduce_mean(tf.square(self.v_lb_pred - self.v_ub_pred)) + \tf.reduce_mean(tf.square(self.u_x_lb_pred - self.u_x_ub_pred)) + \tf.reduce_mean(tf.square(self.v_x_lb_pred - self.v_x_ub_pred)) + \tf.reduce_mean(tf.square(self.f_u_pred)) + \tf.reduce_mean(tf.square(self.f_v_pred))# MSE0|u|^2 + MSE0|v|^2 +# MSEb|u(-5)-u(5)|^2 + MSEb|v(-5)-v(5)|^2 +# MSEb|u_x(-5)-u_x(5)|^2 + MSEb|v_x(-5)-v_x(5)|^2# MSEf|u|^2 + MSEf|v|^2# 获取损失函数历史记录# self.optimizer = tf.contrib.opt.ScipyOptimizerInterface(self.loss,     # 将tf.contrib.opt改为tf.compat.v1.estimator.opt#                                                         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)    # 反向传播算法更新权重和偏置# tf sessionself.sess = tf.Session(config=tf.ConfigProto(allow_soft_placement=True,log_device_placement=True))init = tf.global_variables_initializer()self.sess.run(init)def initialize_NN(self, layers):"""初始化网络权重和偏置参数:param layers: eg.[2, 100, 100, 100, 100, 2]:return:"""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, biasesdef xavier_init(self, size):"""正态分布初始化权重tf.compat.v1.random.truncated_normal(维度，正态分布均值，正态分布标准差):截断的产生正态分布的随机数，即随机数与均值的差值若大于两倍的标准差，则重新生成。:param size::return:"""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)return tf.Variable(tf.compat.v1.random.truncated_normal([in_dim, out_dim], stddev=xavier_stddev), dtype=tf.float32)def neural_net(self, X, weights, biases):""":param X: 输入(x,t):param weights: 模型权重:param biases:  模型偏置:return: 返回(u,v),其中u为真解实部，v为真解虚部"""num_layers = len(weights) + 1H = 2.0*(X - self.lb)/(self.ub - self.lb) - 1.0for 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 Ydef net_uv(self, x, t):X = tf.concat([x,t],1)uv = self.neural_net(X, self.weights, self.biases)u = uv[:,0:1]v = uv[:,1:2]u_x = tf.gradients(u, x)[0]v_x = tf.gradients(v, x)[0]return u, v, u_x, v_xdef net_f_uv(self, x, t):u, v, u_x, v_x = self.net_uv(x,t)u_t = tf.gradients(u, t)[0]u_xx = tf.gradients(u_x, x)[0]v_t = tf.gradients(v, t)[0]v_xx = tf.gradients(v_x, x)[0]f_u = u_t + 0.5*v_xx + (u**2 + v**2)*vf_v = v_t - 0.5*u_xx - (u**2 + v**2)*u   return f_u, f_vdef callback(self, loss):print('Loss:', loss)def train(self, nIter):tf_dict = {self.x0_tf: self.x0, self.t0_tf: self.t0,self.u0_tf: self.u0, self.v0_tf: self.v0,self.x_lb_tf: self.x_lb, self.t_lb_tf: self.t_lb,self.x_ub_tf: self.x_ub, self.t_ub_tf: self.t_ub,self.x_f_tf: self.x_f, self.t_f_tf: self.t_f}start_time = time.time()for it in range(nIter):self.sess.run(self.train_op_Adam, tf_dict)# Printif it % 10 == 0:elapsed = time.time() - start_timeloss_value = self.sess.run(self.loss, tf_dict)print('It: %d, Loss: %.3e, Time: %.2f' % (it, loss_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):""":param X_star: 真解坐标值(x,y):return:"""tf_dict = {self.x0_tf: X_star[:,0:1], self.t0_tf: X_star[:,1:2]}u_star = self.sess.run(self.u0_pred, tf_dict)    # 执行sess.run()时，tensorflow并不是计算了整个图，只是计算了与想要fetch的值相关的部分。u_star = self.sess.run(self.v0_pred, tf_dict)    # 这里u_star、u_star是h(t,x)=u(t,x)+iv(t,x)tf_dict = {self.x_f_tf: X_star[:,0:1], self.t_f_tf: X_star[:,1:2]}f_u_star = self.sess.run(self.f_u_pred, tf_dict)    # 这里f_u_star是u_t + 0.5*v_xx + (u**2 + v**2)*vf_v_star = self.sess.run(self.f_v_pred, tf_dict)    # 这里f_v_star是v_t - 0.5*u_xx - (u**2 + v**2)*ureturn u_star, v_star, f_u_star, f_v_starif __name__ == "__main__": noise = 0.0        # Doman boundslb = np.array([-5.0, 0.0])ub = np.array([5.0, np.pi/2])N0 = 50N_b = 50N_f = 20000layers = [2, 100, 100, 100, 100, 2]data = scipy.io.loadmat('../Data/NLS.mat')t = data['tt'].flatten()[:,None]  # (201,1)x = data['x'].flatten()[:,None]  # (256,1)Exact = data['uu']  # (256,201)Exact_u = np.real(Exact)  # (256,201)Exact_v = np.imag(Exact)  # (256,201)Exact_h = np.sqrt(Exact_u**2 + Exact_v**2)X, T = np.meshgrid(x,t)X_star = np.hstack((X.flatten()[:,None], T.flatten()[:,None]))u_star = Exact_u.T.flatten()[:,None]v_star = Exact_v.T.flatten()[:,None]h_star = Exact_h.T.flatten()[:,None]###########################idx_x = np.random.choice(x.shape[0], N0, replace=False)  # 从0-256中选取N0个整数x0 = x[idx_x,:]  # (50,1)u0 = Exact_u[idx_x,0:1]  # (50,1)v0 = Exact_v[idx_x,0:1]  # (50,1)idx_t = np.random.choice(t.shape[0], N_b, replace=False)tb = t[idx_t,:]  # (50,1)X_f = lb + (ub-lb)*lhs(2, N_f)  # (20000,2)  # lhs(因子数，采样数)"""lhs(因子数，采样数):拉丁超立方采样,若因子数为2，默认取样空间是[0,1]*[0,1]这里通过(-5,0)+(10,pi/2)*lhs(2, N_f)可以改变取样空间"""model = PhysicsInformedNN(x0, u0, v0, tb, X_f, layers, lb, ub)start_time = time.time()                model.train(50000)    elapsed = time.time() - start_time                print('Training time: %.4f' % (elapsed))u_pred, v_pred, f_u_pred, f_v_pred = model.predict(X_star)h_pred = np.sqrt(u_pred**2 + v_pred**2)error_u = np.linalg.norm(u_star-u_pred,2)/np.linalg.norm(u_star,2)    # np.linalg.norm求2范数error_v = np.linalg.norm(v_star-v_pred,2)/np.linalg.norm(v_star,2)error_h = np.linalg.norm(h_star-h_pred,2)/np.linalg.norm(h_star,2)print('Error u: %e' % (error_u))print('Error v: %e' % (error_v))print('Error h: %e' % (error_h))U_pred = griddata(X_star, u_pred.flatten(), (X, T), method='cubic')    # (201,256)V_pred = griddata(X_star, v_pred.flatten(), (X, T), method='cubic')H_pred = griddata(X_star, h_pred.flatten(), (X, T), method='cubic')FU_pred = griddata(X_star, f_u_pred.flatten(), (X, T), method='cubic')FV_pred = griddata(X_star, f_v_pred.flatten(), (X, T), method='cubic')     ################################################################################################### Plotting #####################################################################################################    X0 = np.concatenate((x0, 0*x0), 1) X_lb = np.concatenate((0*tb + lb[0], tb), 1) X_ub = np.concatenate((0*tb + ub[0], tb), 1)X_u_train = np.vstack([X0, X_lb, X_ub])fig, ax = newfig(1.0, 0.9)ax.axis('off')####### Row 0: h(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(H_pred.T, interpolation='nearest', cmap='YlGnBu', extent=[lb[1], ub[1], 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)ax.plot(X_u_train[:,1], X_u_train[:,0], 'kx', label = 'Data (%d points)' % (X_u_train.shape[0]), markersize = 4, clip_on = False)line = np.linspace(x.min(), x.max(), 2)[:,None]ax.plot(t[75]*np.ones((2,1)), line, 'k--', linewidth = 1)ax.plot(t[100]*np.ones((2,1)), line, 'k--', linewidth = 1)ax.plot(t[125]*np.ones((2,1)), line, 'k--', linewidth = 1)    ax.set_xlabel('$t$')ax.set_ylabel('$x$')leg = ax.legend(frameon=False, loc = 'best')
#    plt.setp(leg.get_texts(), color='w')ax.set_title('$|h(t,x)|$', fontsize = 10)####### Row 1: h(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_h[:,75], 'b-', linewidth = 2, label = 'Exact')       ax.plot(x,H_pred[75,:], 'r--', linewidth = 2, label = 'Prediction')ax.set_xlabel('$x$')ax.set_ylabel('$|h(t,x)|$')    ax.set_title('$t = %.2f$' % (t[75]), fontsize = 10)ax.axis('square')ax.set_xlim([-5.1,5.1])ax.set_ylim([-0.1,5.1])ax = plt.subplot(gs1[0, 1])ax.plot(x,Exact_h[:,100], 'b-', linewidth = 2, label = 'Exact')       ax.plot(x,H_pred[100,:], 'r--', linewidth = 2, label = 'Prediction')ax.set_xlabel('$x$')ax.set_ylabel('$|h(t,x)|$')ax.axis('square')ax.set_xlim([-5.1,5.1])ax.set_ylim([-0.1,5.1])ax.set_title('$t = %.2f$' % (t[100]), fontsize = 10)ax.legend(loc='upper center', bbox_to_anchor=(0.5, -0.8), ncol=5, frameon=False)ax = plt.subplot(gs1[0, 2])ax.plot(x,Exact_h[:,125], 'b-', linewidth = 2, label = 'Exact')       ax.plot(x,H_pred[125,:], 'r--', linewidth = 2, label = 'Prediction')ax.set_xlabel('$x$')ax.set_ylabel('$|h(t,x)|$')ax.axis('square')ax.set_xlim([-5.1,5.1])ax.set_ylim([-0.1,5.1])    ax.set_title('$t = %.2f$' % (t[125]), fontsize = 10)# plt.show()savefig('./figures/retest/reNLS')

4. 实验细节及复现结果

  这里使用4层全连接神经网络，输入层和输出层各两个神经元，输入层两个神经元分别代表x,tx,tx,t，输出层两个神经元分别代表u(x,t),v(x,t)u(x,t),v(x,t)u(x,t),v(x,t)，隐藏层每层100个神经元。为了计算误差，作者提供了使用谱方法计算的(256∗201)(256*201)(256∗201)个真解，其中第一维度代表空间xxx，第二维度代表时间ttt. 训练50000次之后输出结果如下：

It: 49990, Loss: 8.158e-05, Time: 0.38
Training time: 1970.0348
Error u: 1.154980e+00
Error v: 0.000000e+00
Error h: 4.879406e-01

  为了对比，下面是训练10000次的结果。

  接下来是作者论文中的训练结果。

参考资料

[1]. Physics-informed machine learning