使用深度神经网络求解偏微分方程

Posted 2023-02-16

【中文标题】使用深度神经网络求解偏微分方程

【英文标题】：solving a partial differential equation using a deep neural network

【发布时间】：2022-01-17 02:15:34

【问题描述】：

该程序在计算偏导数时抛出异常：

AttributeError: 'NoneType' object has no attribute 'op'

我怀疑 PDE 函数有问题，但我不知道如何修复它。 我之所以这样编写 PDE 函数是因为我不想依赖外部包 deepxde。一个有效的实现使用 deepxde 包，如下所示：

def pde(x, y):
    dy_t = dde.grad.jacobian(y, x, j=1)
    dy_xx = dde.grad.hessian(y, x, j=0)

    return (
        dy_t
        - dy_xx*0.3)

应该可以使用直接在张量上操作的梯度函数进行微分，以便将计算集成到反向传播算法中。

import tensorflow as tf
#!pip install --upgrade tensorflow
import deepxde as dde
import numpy as np
# Backend tensorflow.compat.v1 or tensorflow
from deepxde.backend import tf
import time 
import matplotlib.pyplot as plt
t0 = time.time()

# Suppress warnings
import warnings
warnings.filterwarnings("ignore")


# ## Helper function to plot $u(x,t)$



def plot(geom_time,data,name):
    plt.figure()
    plt.scatter(geom_time[:,0], geom_time[:,1],s = 20, c=np.reshape(data, (len(data),)), cmap='jet')
    plt.colorbar()
    plt.xlabel('x domain')
    plt.ylabel('time domain')
    plt.title(name)
    plt.show()



def pde(x, y):
    dy_t = tf.gradients(y, x[:, 1:2])
    dy_x = tf.gradients(y, x[:, 0:1])
    dy_xx= tf.gradients(dy_x, x[:, 0:1])
#
    return (
        dy_t
        - dy_xx*0.3)
    

def func(x):
    return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])


# ## 2. initialization
# Define geometry and time horizons for simulation

geom = dde.geometry.Interval(-1, 1)
timedomain = dde.geometry.TimeDomain(0, 1)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)


# Define boundary conditions and initial conditions



bc = dde.DirichletBC(geomtime, func, lambda _, on_boundary: on_boundary)
ic = dde.IC(geomtime, func, lambda _, on_initial: on_initial)


# Create data points needed for building the model

data = dde.data.TimePDE(
    geomtime,
    pde,
    [bc, ic],
    num_domain=4000,
    num_boundary=2000,
    num_initial=1000,
    solution=func,
    num_test=1000,
)


# ### Steps in building the model 
# 
# - Define the number of layers and neurons per layer in the neural network
# - Define activation function on these layers _tanh_
# - Define function for initial weights for training _Glorot uniform_
# - Train the network using Adam optimizer with learning rate of 0.001



layer_size = [2] + [32] * 3 + [1]

activation = "tanh"
initializer = "Glorot uniform"

net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])


# ## 3. Training
# 

t1 = time.time()

losshistory, train_state = model.train(epochs=3000)

t2 = time.time()
print("training time:",(t2-t1))


# ## 4. Post-training Visualization



dde.postprocessing.plot_loss_history(losshistory)
plt.show()


x_data = np.linspace(-1,1,num = 100)
t_data = np.linspace(0,1,num = 100)
test_x , test_t = np.meshgrid(x_data, t_data)
test_domain = np.vstack((np.ravel(test_x), np.ravel(test_t))).T
predicted_solution = model.predict(test_domain)
residual = model.predict(test_domain,operator=pde)



plot(test_domain,predicted_solution,"predicted solution")


plot(test_domain,residual,"residual")


print("total time")
print(t2-t0)

【问题讨论】：

我注意到您将两个不同的（子）模块导入为tf。这可能是问题的一部分吗？

即使只有一个导入语句，问题仍然存在

【参考方案1】：

我找到了一个在 PDE 函数中使用 Tensorflow 梯度的解决方案，如下所示：

def pde(x, y):
#    dy_t = tf.gradients(y, x[:, 1:2])[0]
    dy_t = tf.gradients(y, x)[0][:, 1:2]
#    dy_x = tf.gradients(y, x[:, 0:1])[0]
    dy_x = tf.gradients(y, x)[0][:, 0:1]
#    dy_xx= tf.gradients(dy_x, x[:, 0:1])[0]
    dy_xx= tf.gradients(dy_x, x)[0][:, 0:1]
#
    return (
        dy_t
        - dy_xx*0.3)
    

def func(x):
    return np.sin(np.pi * x[:, 0:1]) * np.exp(-x[:, 1:])

然而，尚不清楚为什么注释掉的语句会导致之前报告的异常。