白交 发自 凹非寺
转自 | 量子位
神经网络概论
“神经网络”一词很流行，人们通常认为它很难，但其实要简单得多。
神经元——神经网络的基本单元
<section style="padding: 0.5em;line-height: 18px;font-size: 14px;letter-spacing: 0px;font-family: Consolas, Inconsolata, Courier, monospace;border-radius: 0px;background: rgb(29, 31, 33);color: rgb(197, 200, 198);margin-left: 8px;margin-right: 8px;overflow-wrap: normal !important;word-break: normal !important;overflow: auto !important;display: -webkit-box !important;">import numpy as np<br /><br /><span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">def</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">sigmoid</span><span style="line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">(x)</span></span>:<br /> <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># Our activation function: f(x) = 1 / (1 + e^(-x))</span><br /> <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">return</span> <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">1</span> / (<span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">1</span> + np.exp(-x))<br /><br /><span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">class</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">Neuron</span>:</span><br /> <span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">def</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">__init__</span><span style="line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">(<span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">self</span>, weights, bias)</span></span>:<br /> <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">self</span>.weights = weights<br /> <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">self</span>.bias = bias<br /><br /> <span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">def</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">feedforward</span><span style="line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">(<span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">self</span>, inputs)</span></span>:<br /> <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># Weight inputs, add bias, then use the activation function</span><br /> total = np.dot(<span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">self</span>.weights, inputs) + <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">self</span>.bias<br /> <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">return</span> sigmoid(total)<br /><br />weights = np.array([<span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">1</span>]) <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># w1 = 0, w2 = 1</span><br />bias = <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">4</span> <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># b = 4</span><br />n = Neuron(weights, bias)<br /><br />x = np.array([<span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">2</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">3</span>]) <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># x1 = 2, x2 = 3</span><br />print(n.feedforward(x)) <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># 0.9990889488055994</span><br /></section>构建神经网络
<section style="padding: 0.5em;line-height: 18px;font-size: 14px;letter-spacing: 0px;font-family: Consolas, Inconsolata, Courier, monospace;border-radius: 0px;background: rgb(29, 31, 33);color: rgb(197, 200, 198);margin-left: 8px;margin-right: 8px;overflow-wrap: normal !important;word-break: normal !important;overflow: auto !important;display: -webkit-box !important;"><span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">import</span> numpy <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">as</span> np<br /><br /><span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># ... code from previous section here</span><br /><br /><span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">class</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">OurNeuralNetwork</span>:</span><br /> <span style="font-size: inherit;line-height: inherit;color: rgb(181, 189, 104);overflow-wrap: inherit !important;word-break: inherit !important;">'''<br /> A neural network with:<br /> - 2 inputs<br /> - a hidden layer with 2 neurons (h1, h2)<br /> - an output layer with 1 neuron (o1)<br /> Each neuron has the same weights and bias:<br /> - w = [0, 1]<br /> - b = 0<br /> '''</span><br /> <span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">def</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">__init__</span><span style="line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">(self)</span>:</span><br /> weights = np.array([<span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">1</span>])<br /> bias = <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span><br /><br /> <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># The Neuron class here is from the previous section</span><br /> self.h1 = Neuron(weights, bias)<br /> self.h2 = Neuron(weights, bias)<br /> self.o1 = Neuron(weights, bias)<br /><br /> <span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">def</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">feedforward</span><span style="line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">(self, x)</span>:</span><br /> out_h1 = self.h1.feedforward(x)<br /> out_h2 = self.h2.feedforward(x)<br /><br /> <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># The inputs for o1 are the outputs from h1 and h2</span><br /> out_o1 = self.o1.feedforward(np.array([out_h1, out_h2]))<br /><br /> <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">return</span> out_o1<br /><br />network = OurNeuralNetwork()<br />x = np.array([<span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">2</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">3</span>])<br />print(network.feedforward(x)) <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># 0.7216325609518421</span></section>训练神经网路——计算损失函数
<section style="padding: 0.5em;line-height: 18px;font-size: 14px;letter-spacing: 0px;font-family: Consolas, Inconsolata, Courier, monospace;border-radius: 0px;background: rgb(29, 31, 33);color: rgb(197, 200, 198);margin-left: 8px;margin-right: 8px;overflow-wrap: normal !important;word-break: normal !important;overflow: auto !important;display: -webkit-box !important;"><span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">import</span> numpy <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">as</span> np<br /><br /><span style="font-size: inherit;color: inherit;line-height: inherit;overflow-wrap: inherit !important;word-break: inherit !important;"><span style="line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">def</span> <span style="line-height: inherit;color: rgb(129, 162, 190);overflow-wrap: inherit !important;word-break: inherit !important;">mse_loss</span><span style="line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">(y_true, y_pred)</span>:</span><br /> <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># y_true and y_pred are numpy arrays of the same length.</span><br /> <span style="font-size: inherit;line-height: inherit;color: rgb(178, 148, 187);overflow-wrap: inherit !important;word-break: inherit !important;">return</span> ((y_true - y_pred) ** <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">2</span>).mean()<br /><br />y_true = np.array([<span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">1</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">1</span>])<br />y_pred = np.array([<span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>, <span style="font-size: inherit;line-height: inherit;color: rgb(222, 147, 95);overflow-wrap: inherit !important;word-break: inherit !important;">0</span>])<br /><br />print(mse_loss(y_true, y_pred)) <span style="font-size: inherit;line-height: inherit;color: rgb(150, 152, 150);overflow-wrap: inherit !important;word-break: inherit !important;"># 0.5</span><br /></section>训练神经网络——最小化损失
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—完—
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