一、数学基础标量和向量、矩阵、张量tensor矩阵乘法MxN矩阵乘 NxP矩阵得到 MxP维矩阵numpy常用操作.dim #秩 轴的数量或维度的数量 .shape #对象的尺度 .size #对象的个数 .dtype #对象类型 .reshape(shape) #不改变数组元素返回一个shape形状的数组原数组不变 .resize(shape) #与reshape功能一致但改变原数组 .swapaxes(ax1,ax2) #将数组n个维度中两个进行交换 .flatten() #对数组进行降维返回折叠后的一维数组原数组不变 np.arange(n) #类似range函数放回ndarray类型元素0~n-1 np.ones(shape) #根据shape生成一个全1数组 np.zeros(shape) #根据shape生成一个全0数组 np.full(shape,val) #根据shape生成一个全val数组 np.eye(n) #创建一个正方n*n的矩阵对角线全为1其余为0 np.linspace(b,e,n) #根据起始值等间距填充数据形成数组二、导数三、梯度下降算法意义多元函数的导数例原函数则导函数在【111】处的梯度为【6815】梯度是一个向量1.梯度下降法根据梯度更新权重学习率控制权重更新的幅度SGD Stochastic gradient descent2.代码示例选定模型结构def func(x): y w1 * x**2 w2 * x w3 return y模型参数随机初始化w1, w2, w3 1, 0, -1损失函数选用def loss(y_pred, y_true): return (y_pred - y_true) ** 2学习率设置lr 0.1训练过程for epoch in range(1000): epoch_loss 0 for x, y_true in zip(X, Y): # 根据当前模型参数获取预测值 y_pred func(x) # 累加当轮误差值 epoch_loss loss(y_pred, y_true) # 梯度计算 grad_w1 2 * (y_pred - y_true) * x ** 2 grad_w2 2 * (y_pred - y_true) * x grad_w3 2 * (y_pred - y_true) # 根据SGD公式进行权重更新 w1 w1 - lr * grad_w1 #sgd w2 w2 - lr * grad_w2 w3 w3 - lr * grad_w3 epoch_loss / len(X) print(第%d轮 loss %f %(epoch, epoch_loss)) # 训练后误差达到目标值内 if epoch_loss 0.0001: break print(f训练后权重:w1:{w1} w2:{w2} w3:{w3})3.反向传播过程1.根据输入x和模型当前权重计算预测值y2.根据y和y使用loss函数计算loss3.根据loss计算模型权重的梯度4.使用梯度和学习率根据优化器调整模型权重权重更新方式Gradient descent 所有样本一起计算梯度累加Stochastic gradient descent 每次使用一个样本计算梯度Mini-batch gradient descent 每次使用n个样本计算梯度累加import math import sys # X [0.01 * x for x in range(100)] # Y [2*x**2 3*x 4 for x in X] # X [0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19, 0.2, 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.3, 0.31, 0.32, 0.33, 0.34, 0.35000000000000003, 0.36, 0.37, 0.38, 0.39, 0.4, 0.41000000000000003, 0.42, 0.43, 0.44, 0.45, 0.46, 0.47000000000000003, 0.48, 0.49, 0.5, 0.51, 0.52, 0.53, 0.54, 0.55, 0.56, 0.5700000000000001, 0.58, 0.59, 0.6, 0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67, 0.68, 0.6900000000000001, 0.7000000000000001, 0.71, 0.72, 0.73, 0.74, 0.75, 0.76, 0.77, 0.78, 0.79, 0.8, 0.81, 0.8200000000000001, 0.8300000000000001, 0.84, 0.85, # 0.86, 0.87, 0.88, 0.89, 0.9, 0.91, 0.92, 0.93, 0.9400000000000001, 0.9500000000000001, 0.96, 0.97, 0.98, 0.99] # Y [4.0, 4.0302, 4.0608, 4.0918, 4.1232, 4.155, 4.1872, 4.2198, 4.2528, 4.2862, 4.32, 4.3542, 4.3888, 4.4238, 4.4592, 4.495, 4.5312, 4.5678, 4.6048, 4.6422, 4.68, 4.7181999999999995, 4.7568, 4.7958, 4.8352, 4.875, 4.9152000000000005, 4.9558, 4.9968, 5.0382, 5.08, 5.122199999999999, 5.1648, 5.2078, 5.2512, 5.295, 5.3392, 5.3838, 5.4288, 5.4742, 5.5200000000000005, 5.5662, 5.6128, 5.6598, 5.7072, 5.755, 5.8032, 5.851800000000001, 5.9008, 5.9502, 6.0, 6.0502, 6.1008, 6.1518, 6.203200000000001, 6.255000000000001, 6.3072, 6.3598, 6.4128, 6.4662, 6.52, 6.5742, 6.6288, 6.6838, 6.7392, 6.795, 6.8512, 6.9078, 6.9648, 7.022200000000001, 7.08, 7.138199999999999, 7.1968, 7.2558, 7.3152, 7.375, 7.4352, 7.4958, 7.5568, 7.6182, 7.680000000000001, 7.7422, 7.8048, 7.867800000000001, 7.9312, 7.994999999999999, 8.0592, 8.1238, 8.1888, 8.2542, 8.32, 8.3862, 8.4528, 8.5198, 8.587200000000001, 8.655000000000001, 8.7232, 8.7918, 8.8608, 8.9302] X [0.01 * x for x in range(100)] Y [2*x**2 3*x 4 for x in X] def func(x): y w1 * x**2 w2 * x w3 return y def loss(y_pred, y_true): return (y_pred - y_true) ** 2 # 权重随机初始化 w1, w2, w3 1, 0, -1 # 学习率设置 lr 0.1 # 训练过程 for epoch in range(1000): epoch_loss 0 for x, y_true in zip(X, Y): y_pred func(x) epoch_loss loss(y_pred, y_true) #梯度计算 grad_w1 2 * (y_pred - y_true) * x ** 2 grad_w2 2 * (y_pred - y_true) * x grad_w3 2 * (y_pred - y_true) #权重更新 w1 w1 - lr * grad_w1 #sgd w2 w2 - lr * grad_w2 w3 w3 - lr * grad_w3 epoch_loss / len(X) print(第%d轮 loss %f %(epoch, epoch_loss)) if epoch_loss 0.0001: break print(f训练后权重:w1:{w1} w2:{w2} w3:{w3}) #使用训练后模型输出预测值 Yp [func(i) for i in X]四、pytorchhttps://pytorch.org/pytorch官方文档五、常见网络结构1.全连接层线性层计算公式y w * x bw和b是参与训练的参数w的维度决定了隐含层输出的维度一般称为隐单元个数hidden size举例输入x 维度1 x 3 隐含层1w维度3 x 5 隐含层2: w维度5 x 2代码示例模型定义搭建一个2层神经网络模型class TorchModel(nn.Module): def __init__(self, input_size, hidden_size1, hidden_size2): super(TorchModel, self).__init__() self.layer1 nn.Linear(input_size, hidden_size1) #w3 * 5 self.layer2 nn.Linear(hidden_size1, hidden_size2) # 5 * 2 def forward(self, x): x self.layer1(x) #shape: (batch_size, input_size) - (batch_size, hidden_size1) y_pred self.layer2(x) #shape: (batch_size, hidden_size1) - (batch_size, hidden_size2) return y_pred准备输入数据、建立模型x np.array([[3.1, 1.3, 1.2], [2.1, 1.3, 13]]) #建立torch模型 torch_model TorchModel(3, 5, 2)使用torch模型进行预测torch_x torch.FloatTensor(x) y_pred torch_model.forward(torch_x) print(torch模型预测结果, y_pred)2.激活函数为模型添加非线性因素使模型具有拟合非线性函数的能力。无激活函数时 y w1(w2(w3 * x b3) b2) b1 仍然是线性函数2.1.Sigmoid函数2.2.tanh函数2.3.Relu函数2.4.Gelu函数2.5.Softmax函数3.损失函数3.1.均方差MSE mean square error 对均方差在做开根号可以得到根方差3.2.交叉熵Cross Entropy 常用于分类任务 分类任务中网络输出经常是所有类别上的概率分布公式假设一个三分类任务某样本的正确标签是第一类则p [1, 0, 0], 模型预测值假设为[0.5, 0.4, 0.1], 则交叉熵计算如下