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计算图结构

分析：

1. 起始节点 a
2. b = 5 - 3a
3. c = 2b + 3
4. d = 5b + 6
5. e = 7c + d^2
6. f = 2e
7. 最终输出 g = 3f - o（其中 o 是另一个输入）

前向传播

前向传播按照上述顺序计算每个节点的值。

反向传播过程

反向传播的目标是计算损失函数（这里假设为 g）对每个中间变量和输入的偏导数。从右向左进行计算：

1. ∂g/∂o = -1
2. ∂g/∂f = 3
3. ∂f/∂e = 2
4. ∂e/∂c = 7
5. ∂e/∂d = 2d
6. ∂d/∂b = 5
7. ∂c/∂b = 2
8. ∂b/∂a = -3

链式法则应用

使用链式法则计算出 g 对每个变量的全导数：

1. dg/df = ∂g/∂f = 3
2. dg/de = (∂g/∂f) * (∂f/∂e) = 3 * 2 = 6
3. dg/dc = (dg/de) * (∂e/∂c) = 6 * 7 = 42
4. dg/dd = (dg/de) * (∂e/∂d) = 6 * 2d
5. dg/db = (dg/dc) * (∂c/∂b) + (dg/dd) * (∂d/∂b)
   = 42 * 2 + 6 * 2d * 5
   = 84 + 60d
6. dg/da = (dg/db) * (∂b/∂a)
   = (84 + 60d) * (-3)
   = -252 - 180d

最终梯度

最终得到 g 对输入 a 和 o 的梯度：

• dg/da = -252 - 180d
• dg/do = -1

代码实现

静态图

import mathclass Node:"""表示计算图中的一个节点。每个节点都可以存储一个值、梯度，并且知道如何计算前向传播和反向传播。"""def __init__(self, value=None):self.value = value  # 节点的值self.gradient = 0   # 节点的梯度self.parents = []   # 父节点列表self.forward_fn = lambda: None  # 前向传播函数self.backward_fn = lambda: None  # 反向传播函数def __add__(self, other):"""加法操作"""return self._create_binary_operation(other, lambda x, y: x + y, lambda: (1, 1))def __mul__(self, other):"""乘法操作"""return self._create_binary_operation(other, lambda x, y: x * y, lambda: (other.value, self.value))def __sub__(self, other):"""减法操作"""return self._create_binary_operation(other, lambda x, y: x - y, lambda: (1, -1))def __pow__(self, power):"""幂运算"""result = Node()result.parents = [self]def forward():result.value = math.pow(self.value, power)def backward():self.gradient += power * math.pow(self.value, power-1) * result.gradientresult.forward_fn = forwardresult.backward_fn = backwardreturn resultdef _create_binary_operation(self, other, forward_op, gradient_op):"""创建二元操作的辅助方法。用于简化加法、乘法和减法的实现。"""result = Node()result.parents = [self, other]def forward():result.value = forward_op(self.value, other.value)def backward():grads = gradient_op()self.gradient += grads[0] * result.gradientother.gradient += grads[1] * result.gradientresult.forward_fn = forwardresult.backward_fn = backwardreturn resultdef topological_sort(node):"""对计算图进行拓扑排序。确保在前向和反向传播中按正确的顺序处理节点。"""visited = set()topo_order = []def dfs(n):if n not in visited:visited.add(n)for parent in n.parents:dfs(parent)topo_order.append(n)dfs(node)return topo_order# 构建计算图
a = Node(2)  # 假设a的初始值为2
o = Node(1)  # 假设o的初始值为1# 按照给定的数学表达式构建计算图
b = Node(5) - a * Node(3)
c = b * Node(2) + Node(3)
d = b * Node(5) + Node(6)
e = c * Node(7) + d ** 2
f = e * Node(2)
g = f * Node(3) - o# 前向传播
sorted_nodes = topological_sort(g)
for node in sorted_nodes:node.forward_fn()# 反向传播
g.gradient = 1  # 设置输出节点的梯度为1
for node in reversed(sorted_nodes):node.backward_fn()# 打印结果
print(f"g = {g.value}")
print(f"dg/da = {a.gradient}")
print(f"dg/do = {o.gradient}")# 验证手动计算的结果
d_value = 5 * b.value + 6
expected_dg_da = -252 - 180 * d_value
print(f"Expected dg/da = {expected_dg_da}")
print(f"Difference: {abs(a.gradient - expected_dg_da)}")

动态图

import mathclass Node:"""表示计算图中的一个节点。实现了动态计算图的核心功能，包括前向计算和反向传播。"""def __init__(self, value, children=(), op=''):self.value = value  # 节点的值self.grad = 0       # 节点的梯度self._backward = lambda: None  # 反向传播函数，默认为空操作self._prev = set(children)  # 前驱节点集合self._op = op  # 操作符，用于调试def __add__(self, other):"""加法操作"""other = other if isinstance(other, Node) else Node(other)result = Node(self.value + other.value, (self, other), '+')def _backward():self.grad += result.gradother.grad += result.gradresult._backward = _backwardreturn resultdef __mul__(self, other):"""乘法操作"""other = other if isinstance(other, Node) else Node(other)result = Node(self.value * other.value, (self, other), '*')def _backward():self.grad += other.value * result.gradother.grad += self.value * result.gradresult._backward = _backwardreturn resultdef __pow__(self, other):"""幂运算"""assert isinstance(other, (int, float)), "only supporting int/float powers for now"result = Node(self.value ** other, (self,), f'**{other}')def _backward():self.grad += (other * self.value**(other-1)) * result.gradresult._backward = _backwardreturn resultdef __neg__(self):"""取反操作"""return self * -1def __sub__(self, other):"""减法操作"""return self + (-other)def __truediv__(self, other):"""除法操作"""return self * other**-1def __radd__(self, other):"""反向加法"""return self + otherdef __rmul__(self, other):"""反向乘法"""return self * otherdef __rtruediv__(self, other):"""反向除法"""return other * self**-1def tanh(self):"""双曲正切函数"""x = self.valuet = (math.exp(2*x) - 1)/(math.exp(2*x) + 1)result = Node(t, (self,), 'tanh')def _backward():self.grad += (1 - t**2) * result.gradresult._backward = _backwardreturn resultdef backward(self):"""执行反向传播，计算梯度。使用拓扑排序确保正确的反向传播顺序。"""topo = []visited = set()def build_topo(v):if v not in visited:visited.add(v)for child in v._prev:build_topo(child)topo.append(v)build_topo(self)self.grad = 1  # 设置输出节点的梯度为1for node in reversed(topo):node._backward()  # 对每个节点执行反向传播def main():"""主函数，用于测试自动微分系统。构建一个计算图，执行反向传播，并验证结果。"""# 构建计算图a = Node(2)o = Node(1)b = Node(5) - a * 3c = b * 2 + 3d = b * 5 + 6e = c * 7 + d ** 2f = e * 2g = f * 3 - o# 反向传播g.backward()# 打印结果print(f"g = {g.value}")print(f"dg/da = {a.grad}")print(f"dg/do = {o.grad}")# 验证手动计算的结果d_value = 5 * b.value + 6expected_dg_da = -252 - 180 * d_valueprint(f"Expected dg/da = {expected_dg_da}")print(f"Difference: {abs(a.grad - expected_dg_da)}")if __name__ == "__main__":main()

解释:

1. Node 类代表计算图中的一个节点,包含值、梯度、父节点以及前向和反向传播函数。
2. 重载的数学运算符 (__add__, __mul__, __sub__, __pow__) 允许直观地构建计算图。
3. _create_binary_operation 方法用于创建二元操作,简化了加法、乘法和减法的实现。
4. topological_sort 函数对计算图进行拓扑排序,确保正确的计算顺序。

import mathclass Node:"""表示计算图中的一个节点。实现了动态计算图的核心功能，包括前向计算和反向传播。"""def __init__(self, value, children=(), op=''):self.value = value  # 节点的值self.grad = 0       # 节点的梯度self._backward = lambda: None  # 反向传播函数，默认为空操作self._prev = set(children)  # 前驱节点集合self._op = op  # 操作符，用于调试def __add__(self, other):"""加法操作"""other = other if isinstance(other, Node) else Node(other)result = Node(self.value + other.value, (self, other), '+')def _backward():self.grad += result.gradother.grad += result.gradresult._backward = _backwardreturn resultdef __mul__(self, other):"""乘法操作"""other = other if isinstance(other, Node) else Node(other)result = Node(self.value * other.value, (self, other), '*')def _backward():self.grad += other.value * result.gradother.grad += self.value * result.gradresult._backward = _backwardreturn resultdef __pow__(self, other):"""幂运算"""assert isinstance(other, (int, float)), "only supporting int/float powers for now"result = Node(self.value ** other, (self,), f'**{other}')def _backward():self.grad += (other * self.value**(other-1)) * result.gradresult._backward = _backwardreturn resultdef __neg__(self):"""取反操作"""return self * -1def __sub__(self, other):"""减法操作"""return self + (-other)def __truediv__(self, other):"""除法操作"""return self * other**-1def __radd__(self, other):"""反向加法"""return self + otherdef __rmul__(self, other):"""反向乘法"""return self * otherdef __rtruediv__(self, other):"""反向除法"""return other * self**-1def tanh(self):"""双曲正切函数"""x = self.valuet = (math.exp(2*x) - 1)/(math.exp(2*x) + 1)result = Node(t, (self,), 'tanh')def _backward():self.grad += (1 - t**2) * result.gradresult._backward = _backwardreturn resultdef backward(self):"""执行反向传播，计算梯度。使用拓扑排序确保正确的反向传播顺序。"""topo = []visited = set()def build_topo(v):if v not in visited:visited.add(v)for child in v._prev:build_topo(child)topo.append(v)build_topo(self)self.grad = 1  # 设置输出节点的梯度为1for node in reversed(topo):node._backward()  # 对每个节点执行反向传播def main():"""主函数，用于测试自动微分系统。构建一个计算图，执行反向传播，并验证结果。"""# 构建计算图a = Node(2)o = Node(1)b = Node(5) - a * 3c = b * 2 + 3d = b * 5 + 6e = c * 7 + d ** 2f = e * 2g = f * 3 - o# 反向传播g.backward()# 打印结果print(f"g = {g.value}")print(f"dg/da = {a.grad}")print(f"dg/do = {o.grad}")# 验证手动计算的结果d_value = 5 * b.value + 6expected_dg_da = -252 - 180 * d_valueprint(f"Expected dg/da = {expected_dg_da}")print(f"Difference: {abs(a.grad - expected_dg_da)}")if __name__ == "__main__":main()

解释：

1. Node 类是核心，它代表计算图中的一个节点，并实现了各种数学运算。

2. 每个数学运算（如 __add__, __mul__ 等）都创建一个新的 Node，并定义了相应的反向传播函数。

3. backward 方法实现了反向传播算法，使用拓扑排序确保正确的计算顺序。