如何用神经网络近似函数

如何解决如何用神经网络近似函数

学习神经网络我已经编写了自己的课程。

import numpy as np
import random


def sigmoid(x):
    return 1.0 / (1.0 + np.exp(-x))


def sigmoid_prime(x):
    return sigmoid(x) * (1 - sigmoid(x))


def linear(x):
    return x

def linear_prime(x):
    return 1

def tanh(x):
    return (np.exp(x) - np.exp(-x))/(np.exp(x) + np.exp(-x))

def tanh_prime(x):
    return 1 - tanh(x)*tanh(x)

class Network:
    def __init__(self,sizes,activation_func = sigmoid,activation_prime = sigmoid_prime):
        
        self.biases = [np.random.randn(x,1) for x in sizes[1:]]
        self.weights = [np.random.randn(y,x) for x,y in zip(sizes,sizes[1:])]

        self.num_layers = len(sizes)
        self.sizes = sizes

        self.activation_function = activation_func
        self.actiovation_prime = activation_prime

    def forward_prop(self,a):
        for w,b in zip(self.weights,self.biases):
            a = self.activation_function(np.dot(w,a) + b)
        return a

    def cost_derivative(self,output_activations,y):
        return (output_activations - y)

    def backprop(self,x,y):

        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]

        # forward pass

        activation = x  # first activation,which is input layer
        a_mas = [x]
        z_mas = []

        for b,w in zip(self.biases,self.weights):
            z = np.dot(w,activation) + b
            activation = self.activation_function(z)
            z_mas.append(z)
            a_mas.append(activation)
            pass

        # backward pass
        
        delta = self.cost_derivative(a_mas[-1],y) * self.actiovation_prime(z_mas[-1])
        nabla_b[-1] = delta
        nabla_w[-1] = np.dot(delta,a_mas[-2].T)

        for l in range(2,self.num_layers):  # there is 2 such as we've already done for last layer

            delta = np.dot(self.weights[-l + 1].transpose(),delta) * self.actiovation_prime(z_mas[-l])
            nabla_b[-l] = delta
            nabla_w[-l] = np.dot(delta,a_mas[-l - 1].T)

        return nabla_b,nabla_w

    def update_mini_batch(self,mini_batch,eta):

        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]

        for x,y in mini_batch:
            delta_nabla_b,delta_nabla_w = self.backprop(x,y)
            nabla_b = [nb + dnb for nb,dnb in zip(nabla_b,delta_nabla_b)]
            nabla_w = [nw + dnw for nw,dnw in zip(nabla_w,delta_nabla_w)]

        eps = eta / len(mini_batch)
        self.weights = [w - eps * nw for w,nw in zip(self.weights,nabla_w)]
        self.biases = [b - eps * nb for b,nb in zip(self.biases,nabla_b)]

    def SGD(self,training_data,epochs,mini_batch_size,eta):
        n = len(training_data)

        for j in range(epochs):
            random.shuffle(training_data)
            mini_batches = [training_data[k:k + mini_batch_size]
                            for k in range(0,n,mini_batch_size)]

            for mini_batch in mini_batches:
                self.update_mini_batch(mini_batch,eta)

现在我正试图借助该网络来近似sin（）。但是下面的以下代码无法正常工作。

%matplotlib inline
import matplotlib.pyplot as plt

net2 = Network([1,100,1],tanh,tanh_prime)

x = np.linspace(0,10,1000)
y = np.sin(x)



train = [(np.array(x[i]).reshape(1,1),np.array(y[i]).reshape(1,1)) for i in range(len(x))]
    
    
net2.SGD(train,0.1)

y_pred = []
y_tmp = []

for i in range(len(x)):
    y_tmp.append(net2.forward_prop(train[i][0]))
    y_pred.append(float(net2.forward_prop(train[i][0])))


plt.plot(x,y_look,'r',y_pred)
plt.grid()

这就是我得到的。

我已经尝试借助MNIST数据集在数字识别中实现此网络。那里一切正常。但是我无法获得比70％更高的准确度，但这不是问题。但是这里我不知道出了什么问题。激活函数是tanh（）。

解决方法

据我所知，您正在使函数subset(df,!PID %in% PID[Stage >= 4]) 最小化，您可能希望将其更改为均方误差（f(x) - y）或均绝对误差（(f(x) - y)^2），分别为足以解决像您这样的回归问题。对于诸如MNIST之类的分类问题，交叉熵是一个不错的选择。

此外，您可能会尝试在输出层中删除tanh函数。我不认为|f(x) - y|是个问题，因为输出在-1和1之间，但是通常，对于回归问题，我们使用线性激活，对于分类问题，我们保留诸如Sigmoid和tanh的压扁函数。