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|
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|
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函数逼近
|
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优化
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概率论
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概率概念
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分布
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</a>
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贝叶斯
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</a>
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信息论
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机器学习
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机器学习
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经典机器学习
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梯度机器学习
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深度学习
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强化学习
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分布式深度学习
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</span>
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|
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计算语言学
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语言学基础
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嵌入与序列模型
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Transformer 与语言模型
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</span>
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</a>
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高级文本生成
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</span>
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|
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计算机视觉
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图像基础
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目标检测与分割
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音频与语音
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数字信号处理
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自动语音识别
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语音合成
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</span>
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</a>
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<span class="md-ellipsis">
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说话人与音频分析
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</span>
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|
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|
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</a>
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</li>
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源分离与降噪
|
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</span>
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|
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<span class="md-ellipsis">
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多模态学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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多模态学习
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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图像与视频 Token 化
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</span>
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|
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|
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</a>
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跨模态生成
|
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
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统一多模态架构
|
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|
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|
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</span>
|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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自主系统
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</span>
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<span class="md-nav__icon md-icon"></span>
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感知
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</a>
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机器人学习
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</span>
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</a>
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视觉-语言-动作模型
|
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</span>
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|
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|
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|
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</a>
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自动驾驶
|
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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|
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|
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|
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</span>
|
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|
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|
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|
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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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几何深度学习
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</span>
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硬件基础
|
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|
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</span>
|
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|
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</a>
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ARM 与 NEON
|
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x86 与 AVX
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Triton、TPU 与 Pallas
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ML 系统设计
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应用 AI
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前沿 AI
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神经形态计算
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太空数据中心
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去中心化 AI
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编程练习（使用 CoLab 或 notebook）
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<h1 id="_1">微分<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>微分学研究瞬时变化率。本节涵盖极限、导数、微分法则、链式法则（反向传播的基础），以及机器学习中常用的导数。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>在前面的章节中，我们学会了如何将数据表示为向量，并用矩阵对其进行变换。但现实世界中的许多现象并非静止不变的。汽车在加速，股价在波动，神经网络的损失随着权重的更新而变化。<strong>微积分</strong>是研究变化的数学。</p>
|
||
</li>
|
||
<li>
|
||
<p>微积分回答两个问题：某个量在当前时刻变化得有多快？（微分学）以及它在一段时间内累积了多少？（积分学）。本节讨论的是"多快"的问题。</p>
|
||
</li>
|
||
<li>
|
||
<p>想象一下你正在开车，瞥了一眼车速表。上面显示 60 km/h。这个数字不是你整个行程的平均速度，而是你在这一瞬间的瞬时速度。微分学为我们提供了计算这种瞬时变化率的工具。</p>
|
||
</li>
|
||
<li>
|
||
<p>但首先，让我们回顾一下直线方程：<span class="arithmatex">\(y = mx + b\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>这是两个量之间最简单的关系。</p>
|
||
<ul>
|
||
<li><span class="arithmatex">\(b\)</span> 是 <strong>y 截距</strong>，即直线与 y 轴的交点（当 <span class="arithmatex">\(x = 0\)</span> 时的起始值）。</li>
|
||
<li><span class="arithmatex">\(m\)</span> 是<strong>斜率</strong>，即变化率：<span class="arithmatex">\(x\)</span> 每增加 1 个单位，<span class="arithmatex">\(y\)</span> 就变化 <span class="arithmatex">\(m\)</span> 个单位。</li>
|
||
<li>如果 <span class="arithmatex">\(m = 3\)</span>，直线陡峭上升；如果 <span class="arithmatex">\(m = 0\)</span>，直线水平；如果 <span class="arithmatex">\(m = -2\)</span>，直线下降。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>斜率计算公式为 <span class="arithmatex">\(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\)</span>，即"<span class="arithmatex">\(y\)</span> 变化了多少"与"<span class="arithmatex">\(x\)</span> 变化了多少"的比值。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="直线方程：b 是 y 截距，m 是斜率（纵坐标变化除以横坐标变化）" src="../../images/line_equation.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>一旦知道了 <span class="arithmatex">\(m\)</span> 和 <span class="arithmatex">\(b\)</span>，就可以计算任意 <span class="arithmatex">\(x\)</span> 对应的 <span class="arithmatex">\(y\)</span> 值。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，若 <span class="arithmatex">\(m = 2\)</span> 且 <span class="arithmatex">\(b = 3\)</span>，则在 <span class="arithmatex">\(x = 5\)</span> 处：<span class="arithmatex">\(y = 2(5) + 3 = 13\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>这两个参数完全决定了这条直线，预测任何输出只需代入公式即可。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于直线，斜率处处相同。</p>
|
||
</li>
|
||
<li>
|
||
<p>这一思想可以推广到直线之外。任何函数都是一个将输入映射到输出的规则，一旦知道了它的公式（参数和形状），就可以计算任意输入对应的输出，并将结果绘制成图。</p>
|
||
</li>
|
||
<li>
|
||
<p><span class="arithmatex">\(y = x^2\)</span> 给出抛物线，<span class="arithmatex">\(y = \sin(x)\)</span> 给出波形，<span class="arithmatex">\(y = e^x\)</span> 给出指数增长。每个公式都定义了一条特定的曲线，能够熟练地将函数理解为一种形状，对于后续内容至关重要。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于直线，斜率处处相同。但大多数有趣的函数都是弯曲的，因此斜率在不同点处各不相同。微积分给了我们一种方法来求曲线上任意一点的斜率。</p>
|
||
</li>
|
||
<li>
|
||
<p>我们还需要<strong>极限</strong>的概念。极限描述的是当输入越来越接近某个目标值时，函数趋近于什么值，而不一定非要达到该值。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\lim_{x \to a} f(x) = L\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这读作："当 <span class="arithmatex">\(x\)</span> 趋近于 <span class="arithmatex">\(a\)</span> 时，<span class="arithmatex">\(f(x)\)</span> 趋近于 <span class="arithmatex">\(L\)</span>。"函数在 <span class="arithmatex">\(x = a\)</span> 处不一定等于 <span class="arithmatex">\(L\)</span>，只需无限接近即可。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，考虑 <span class="arithmatex">\(f(x) = \frac{x^2 - 1}{x - 1}\)</span>。如果直接代入 <span class="arithmatex">\(x = 1\)</span>，会得到 <span class="arithmatex">\(\frac{0}{0}\)</span>，这是未定义的。</p>
|
||
</li>
|
||
<li>
|
||
<p>但尝试接近 1 的值：<span class="arithmatex">\(f(0.9) = 1.9\)</span>，<span class="arithmatex">\(f(0.99) = 1.99\)</span>，<span class="arithmatex">\(f(1.01) = 2.01\)</span>。输出显然朝着 2 靠近。</p>
|
||
</li>
|
||
<li>
|
||
<p>从代数角度看，我们可以理解原因：将分子因式分解为 <span class="arithmatex">\((x-1)(x+1)\)</span>，约去 <span class="arithmatex">\((x-1)\)</span> 项，对于所有 <span class="arithmatex">\(x \neq 1\)</span> 得到 <span class="arithmatex">\(f(x) = x + 1\)</span>。因此当 <span class="arithmatex">\(x \to 1\)</span> 时，<span class="arithmatex">\(f(x) \to 2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>该函数在 <span class="arithmatex">\(x = 1\)</span> 处有一个空洞，但极限仍然存在。</p>
|
||
</li>
|
||
<li>
|
||
<p>极限是微积分中其他一切内容的基础。</p>
|
||
</li>
|
||
<li>
|
||
<p>函数 <span class="arithmatex">\(f(x)\)</span> 在点 <span class="arithmatex">\(x = a\)</span> 处的<strong>导数</strong>衡量的是瞬时变化率。从几何角度看，它是该点处曲线切线的斜率。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="导数就是曲线上某点处切线的斜率" src="../../images/tangent_line.svg" /></p>
|
||
<ul>
|
||
<li>要计算这个斜率，我们首先在曲线上取两个点，计算通过这两个点的直线（<strong>割线</strong>）的斜率。然后让第二个点逐渐靠近第一个点，观察割线的斜率趋近于什么值。这就是<strong>差商</strong>：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]</div>
|
||
<p><img alt="随着 h 趋近于 0，割线趋近于切线" src="../../images/difference_quotient.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>分子 <span class="arithmatex">\(f(a+h) - f(a)\)</span> 是输出的变化量。分母 <span class="arithmatex">\(h\)</span> 是输入的变化量。它们的比值是在一个极小区间上的平均变化率。当 <span class="arithmatex">\(h \to 0\)</span> 时，这个平均值就变成了瞬时变化率。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，设 <span class="arithmatex">\(f(x) = x^2\)</span>。在 <span class="arithmatex">\(x = 3\)</span> 处：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f'(3) = \lim_{h \to 0} \frac{(3+h)^2 - 9}{h} = \lim_{h \to 0} \frac{9 + 6h + h^2 - 9}{h} = \lim_{h \to 0} (6 + h) = 6\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>因此在 <span class="arithmatex">\(x = 3\)</span> 处，函数 <span class="arithmatex">\(x^2\)</span> 以每单位输入变化 6 单位输出的速率增加。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果这个极限存在，则称函数在该点是<strong>可微</strong>的。要做到这一点，函数必须连续（没有跳跃）、光滑（没有尖角），并且在点附近有定义。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果你能笔不离纸地画出曲线，且没有任何折点，那么它在该点很可能是可微的。</p>
|
||
</li>
|
||
<li>
|
||
<p>每次都从极限定义出发计算导数会很繁琐。幸运的是，少数几条法则就能让我们快速微分几乎任何函数。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>常数法则</strong>：常数的导数为零。若 <span class="arithmatex">\(f(x) = 5\)</span>，则 <span class="arithmatex">\(f'(x) = 0\)</span>。水平线的斜率为零。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>幂法则</strong>：微分的主力法则。将指数提到前面，然后将指数减一：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{d}{dx} x^n = n x^{n-1}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>例如：<span class="arithmatex">\(\frac{d}{dx} x^3 = 3x^2\)</span>。三次函数变成了二次函数。该法则适用于任何实数指数，包括负数和分数：<span class="arithmatex">\(\frac{d}{dx} x^{-1} = -x^{-2}\)</span> 以及 <span class="arithmatex">\(\frac{d}{dx} \sqrt{x} = \frac{d}{dx} x^{1/2} = \frac{1}{2}x^{-1/2}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>和/差法则</strong>：逐项求导。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\]</div>
|
||
<ul>
|
||
<li><strong>乘积法则</strong>：当两个函数相乘时，导数并非简单地将各自的导数相乘。而是：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>可以这样理解："第一个的变化率乘以第二个，加上第一个乘以第二个的变化率。"例如，<span class="arithmatex">\(\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>商法则</strong>：对于函数的比值：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>一个有用的记忆口诀："上导下不导减去上不导下导，除以分母的平方。"</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>链式法则</strong>：对机器学习最重要的法则。当函数复合（一个函数嵌套在另一个函数内部）时，导数等于沿链各导数的乘积：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\]</div>
|
||
<ul>
|
||
<li>可以把它想象成剥洋葱。先对外层函数求导（内层函数保持不变），然后乘以内层函数的导数。</li>
|
||
</ul>
|
||
<p><img alt="链式法则：对外层求导，乘以内层的导数" src="../../images/chain_rule.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>例如，<span class="arithmatex">\(\frac{d}{dx} (3x + 1)^5 = 5(3x+1)^4 \cdot 3 = 15(3x+1)^4\)</span>。外层函数是 <span class="arithmatex">\((\cdot)^5\)</span>，内层是 <span class="arithmatex">\(3x+1\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>链式法则是神经网络中<strong>反向传播</strong>的数学基础。一个深层网络就是一个由多个复合函数组成的长链。要计算损失相对于每个权重的变化率，我们从输出层开始逐层向输入层反复应用链式法则，将每一步的局部导数相乘。</p>
|
||
</li>
|
||
<li>
|
||
<p>以下是你会遇到的最常见导数。每一个都可以从极限定义推导出来，但熟记它们可以节省时间：</p>
|
||
</li>
|
||
</ul>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>函数</th>
|
||
<th>导数</th>
|
||
<th>备注</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td><span class="arithmatex">\(e^x\)</span></td>
|
||
<td><span class="arithmatex">\(e^x\)</span></td>
|
||
<td>唯一一个导数等于自身的函数</td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(a^x\)</span></td>
|
||
<td><span class="arithmatex">\(a^x \ln a\)</span></td>
|
||
<td>指数函数的一般形式</td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(\ln x\)</span></td>
|
||
<td><span class="arithmatex">\(\frac{1}{x}\)</span></td>
|
||
<td>自然对数</td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(\log_a x\)</span></td>
|
||
<td><span class="arithmatex">\(\frac{1}{x \ln a}\)</span></td>
|
||
<td>一般对数</td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(\sin x\)</span></td>
|
||
<td><span class="arithmatex">\(\cos x\)</span></td>
|
||
<td></td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(\cos x\)</span></td>
|
||
<td><span class="arithmatex">\(-\sin x\)</span></td>
|
||
<td>注意负号</td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(\tan x\)</span></td>
|
||
<td><span class="arithmatex">\(\sec^2 x\)</span></td>
|
||
<td></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<ul>
|
||
<li>
|
||
<p>指数函数 <span class="arithmatex">\(e^x\)</span> 非常特别：它是唯一一个导数等于自身的函数。这就是为什么 <span class="arithmatex">\(e\)</span> 在机器学习中无处不在，从 softmax 激活函数到概率分布都能见到它的身影。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>洛必达法则</strong>用于处理形如 <span class="arithmatex">\(\frac{0}{0}\)</span> 或 <span class="arithmatex">\(\frac{\infty}{\infty}\)</span> 的不定式极限。当直接代入得到这类形式时，可以分别对分子和分母求导，然后再次尝试求极限：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>条件：<span class="arithmatex">\(f\)</span> 和 <span class="arithmatex">\(g\)</span> 都必须在 <span class="arithmatex">\(a\)</span> 附近可微，并且 <span class="arithmatex">\(g'(x)\)</span> 在 <span class="arithmatex">\(a\)</span> 附近（可能除去 <span class="arithmatex">\(a\)</span> 本身）不为零。原极限必须是不定式。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如：<span class="arithmatex">\(\lim_{x \to 0} \frac{\sin x}{x}\)</span>。直接代入得到 <span class="arithmatex">\(\frac{0}{0}\)</span>。应用洛必达法则：<span class="arithmatex">\(\lim_{x \to 0} \frac{\cos x}{1} = 1\)</span>。这个极限是基础的，在信号处理和傅里叶分析中都会出现。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果结果仍然是不定式，可以反复应用该法则。例如，<span class="arithmatex">\(\lim_{x \to 0} \frac{1 - \cos x}{x^2}\)</span> 得到 <span class="arithmatex">\(\frac{0}{0}\)</span>。第一次应用：<span class="arithmatex">\(\lim_{x \to 0} \frac{\sin x}{2x}\)</span>，仍然是 <span class="arithmatex">\(\frac{0}{0}\)</span>。第二次应用：<span class="arithmatex">\(\lim_{x \to 0} \frac{\cos x}{2} = \frac{1}{2}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果两个函数是可微的，那么它们的和、差、积、复合以及商（分母不为零时）也是可微的。这就是为什么我们可以自信地对由简单部分组成的复杂表达式进行微分。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>可视化常见函数。在同一张图中绘制 <span class="arithmatex">\(x^2\)</span>、<span class="arithmatex">\(\sin(x)\)</span> 和 <span class="arithmatex">\(e^x\)</span>，建立对不同公式产生不同形状的直观感受。尝试修改参数（例如 <span class="arithmatex">\(2x^2\)</span>、<span class="arithmatex">\(\sin(2x)\)</span>），观察曲线如何变化。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#e74c3c"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"x² (抛物线)"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#3498db"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"sin(x) (波形)"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">jnp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#27ae60"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"eˣ (指数函数)"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="k">for</span> <span class="n">ax</span> <span class="ow">in</span> <span class="n">axes</span><span class="p">:</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="n">ax</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"gray"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="n">ax</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"gray"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>使用 JAX 的自动微分计算 <span class="arithmatex">\(f(x) = x^3 - 2x + 1\)</span> 在若干点处的导数，并与解析导数 <span class="arithmatex">\(f'(x) = 3x^2 - 2\)</span> 进行比较。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">df</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">]:</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"x=</span><span class="si">{</span><span class="n">x</span><span class="si">:</span><span class="s2">5.1f</span><span class="si">}</span><span class="s2"> 自动微分: </span><span class="si">{</span><span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> 解析解: </span><span class="si">{</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>数值验证链式法则。定义 <span class="arithmatex">\(f(x) = \sin(x^2)\)</span>，通过 <code>jax.grad</code> 计算其导数，并与解析结果 <span class="arithmatex">\(2x\cos(x^2)\)</span> 进行比较。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">df</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">]:</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="n">auto</span> <span class="o">=</span> <span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="n">analytical</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"x=</span><span class="si">{</span><span class="n">x</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2"> 自动微分: </span><span class="si">{</span><span class="n">auto</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2"> 解析解: </span><span class="si">{</span><span class="n">analytical</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>可视化导数。将 <span class="arithmatex">\(f(x) = x^3 - 3x\)</span> 与其导数 <span class="arithmatex">\(f'(x) = 3x^2 - 3\)</span> 绘制在同一张图上。观察 <span class="arithmatex">\(f'(x) = 0\)</span> 的位置与 <span class="arithmatex">\(f\)</span> 的峰谷之间的对应关系。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span>
|
||
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="c1"># jax.grad 用于标量；jax.vmap 将其向量化，可同时处理一组输入</span>
|
||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="n">df</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">vmap</span><span class="p">(</span><span class="n">jax</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">f</span><span class="p">))</span>
|
||
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a>
|
||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
|
||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">jax</span><span class="o">.</span><span class="n">vmap</span><span class="p">(</span><span class="n">f</span><span class="p">)(</span><span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">"f(x)"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">"f'(x)"</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">"--"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"gray"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
|
||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"函数及其导数"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
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