深度学习笔记1：自动微分与神经网络实现(附代码)

import numpy as npclass Variable:def __init__(self, data, grad=None):self.data = dataself.grad = grad if grad is not None else np.zeros_like(data)self.creator = None  # 用于记录创建该变量的函数

class Function:@staticmethoddef forward(x):# 前向传播逻辑raise NotImplementedError@staticmethoddef backward(x, gy):# 反向传播逻辑raise NotImplementedError

#通过继承 Function 类并实现其方法来创建具体的函数，如 Exp 函数
​
class Exp(Function):@staticmethoddef forward(x):return np.exp(x)
​@staticmethoddef backward(x, gy):return gy * np.exp(x)

class Exp(Function):@staticmethoddef forward(x):return np.exp(x)@staticmethoddef backward(x, gy):return gy * np.exp(x)

# 3.2 函数的连续调用
# 通过链式调用多个函数，可以构建复杂的计算图。class Square(Function):
    @staticmethoddef forward(x):return x ** 2    @staticmethoddef backward(x, gy):return 2 * x * gyclass Exp(Function):
    @staticmethoddef forward(x):return np.exp(x)    @staticmethoddef backward(x, gy):return np.exp(x) * gy# 连续调用示例
def function_chain_call():"""此函数展示了函数的连续调用"""x = Variable(np.array(2.0))y = Square.forward(x)z = Exp.forward(y)print(z.data)

def numerical_derivative(f, x, epsilon=1e-4):"""数值微分函数，通过有限差分法近似计算导数参数：f: 要计算导数的函数x: 计算导数的点epsilon: 微小增量，默认为 1e-4返回：函数在 x 点的近似导数"""return (f(x + epsilon) - f(x - epsilon)) / (2 * epsilon)# 示例函数
def example_function(x):return x ** 2 + 3 * x + 1# 计算数值导数
def numerical_derivative_example():"""此函数计算示例函数在特定点的数值导数"""x = 5derivative = numerical_derivative(example_function, x)print(derivative)

# 6.1 Variable 类的功能扩展
class Variable:def __init__(self, data, grad=None, creator=None):self.data = dataself.grad = grad if grad is not None else np.zeros_like(data)self.creator = creator

# 6.2 Function 类的功能扩展
class Function:def __init__(self):self.inputs = []def forward(self, *inputs):self.inputs = inputs# 前向传播逻辑raise NotImplementedErrordef backward(self, gy):# 反向传播逻辑raise NotImplementedError

# 6.3 Square 类和 Exp 类的功能扩展
class Square(Function):
    @staticmethoddef forward(x):y = x.data ** 2output = Variable(y)output.creator = Squarereturn output    @staticmethoddef backward(x, gy):gx = 2 * x.data * gy.datareturn Variable(gx)class Exp(Function):
    @staticmethoddef forward(x):y = np.exp(x.data)output = Variable(y)output.creator = Expreturn output    @staticmethoddef backward(x, gy):gx = np.exp(x.data) * gy.datareturn Variable(gx)

# 6.4 反向传播的实现
def backward(v):"""反向传播函数参数：v: 最终的输出变量"""funcs = []while v.creator is not None:funcs.append(v.creator)v = v.creator.inputs[0]gy = Variable(np.ones_like(v.data))for func in funcs[::-1]:gy = func.backward(func.inputs[0], gy)

# 7.3 增加 backward 方法
class Variable:def backward(self):"""触发反向传播"""backward(self)

# 9.2 简化 backward 方法
class Variable:def backward(self):"""简化 backward 方法的调用"""backward(self)

# 9.1 作为 Python 函数使用
""" 
使 Function 类能够像函数一样被调用
参数：*inputs: 输入参数
返回：计算结果
"""class Function:def __call__(self, *inputs):outputs = self.forward(*inputs)return outputs

# 9.2 简化 backward 方法
class Variable:def backward(self):"""简化 backward 方法的调用"""backward(self)

# 10.1 Python 的单元测试
import unittestclass TestFunctions(unittest.TestCase):def test_square_forward(self):"""测试 Square 函数的前向传播"""x = Variable(np.array(2.0))y = Square.forward(x)self.assertEqual(y.data, 4.0)def test_square_backward(self):"""测试 Square 函数的反向传播"""x = Variable(np.array(2.0))y = Square.forward(x)y.backward()self.assertEqual(x.grad, 4.0)if __name__ == '__main__':unittest.main()

import numpy as np
import unittestclass Square(Function):
    @staticmethoddef forward(x):return x ** 2    @staticmethoddef backward(x, gy):return 2 * x * gyclass TestSquareBackpropagation(unittest.TestCase):def test_square_backpropagation(self):x = Variable(np.array(3.0))y = Square.forward(x)y.backward()expected_grad = 6.0  self.assertAlmostEqual(x.grad, expected_grad, places=5)if __name__ == '__main__':unittest.main()

import numpy as npclass Variable:def __init__(self, data, grad=None):self.data = dataself.grad = grad if grad is not None else np.zeros_like(data)class Function:def forward(self, x):raise NotImplementedErrordef backward(self, x, gy):raise NotImplementedErrorclass Square(Function):def forward(self, x):return x ** 2def backward(self, x, gy):return 2 * x * gydef gradient_check(f, x, epsilon=1e-4):x.grad = Nonef(x).backward()analytic_grad = x.gradnum_grad = np.zeros_like(x.data)for i in range(x.data.size):tmp_val = x.data[i]x.data[i] = tmp_val + epsilonf_pos = f(x).datax.data[i] = tmp_val - epsilonf_neg = f(x).datax.data[i] = tmp_valnum_grad[i] = (f_pos - f_neg) / (2 * epsilon)return np.allclose(analytic_grad, num_grad, atol=1e-5)x = Variable(np.random.rand(5))
if gradient_check(Square, x):print("梯度检验通过")
else:print("梯度检验未通过")

import numpy as np# 假设之前的测试结果存储在一个字典中
test_results = {"square_backpropagation": True,"gradient_check": True
}# 总结测试通过情况
passed_tests = [test_name for test_name, passed in test_results.items() if passed]
failed_tests = [test_name for test_name, passed in test_results.items() if not passed]# 打印通过和未通过的测试
if passed_tests:print("通过的测试：")for test in passed_tests:print(f"- {test}")if failed_tests:print("未通过的测试：")for test in failed_tests:print(f"- {test}")# 分析可能存在的问题和提出改进方向
print("可能的问题：")
if failed_tests:print(" - 未通过测试的函数实现可能有误或未处理好边界情况。")
else:print(" - 复杂场景和极端输入下可能有潜在问题，测试用例覆盖可能不全。")print("改进方向：")
print(" - 调试未通过测试的函数。")
print(" - 补充更多样化的测试用例。")
print(" - 优化数值计算方法和算法。")
print(" - 定期重构和优化代码。")