TensorFlow中高效实现多项式回归：从深度网络到特征工程

import tensorflow as tf
from tensorflow.keras import layers, models

def PolynomialModel_Complex():
    inp = layers.Input((1))
    l = layers.Dense(16, activation='tanh')(inp)
    l = layers.Dense(8, activation='tanh')(l)
    l = layers.Dropout(.5)(l)
    l = layers.Dense(4, activation='tanh')(l)
    l = layers.Dropout(.5)(l)
    output = layers.Dense(1, activation='tanh')(l) # 注意这里的tanh激活函数
    return models.Model(inp, output)

# 假设要拟合 y = 10x
# model_complex = PolynomialModel_Complex()
# model_complex.compile(loss='mean_squared_error', optimizer='adam')
# x_data = tf.linspace(-10, 10, 1000)
# y_data = 10.0 * x_data
# model_complex.fit(x_data, y_data, epochs=100) # 可能会观察到高损失

import tensorflow as tf
from tensorflow.keras import layers, models, optimizers

def PolynomialModel(degree):
    """
    构建一个用于多项式回归的简单Keras模型。
    输入层大小为 degree + 1，对应 [x^0, x^1, ..., x^degree]
    输出层为单个神经元，使用线性激活。
    """
    inp = layers.Input((degree + 1)) # 输入大小为 degree + 1
    out = layers.Dense(1, activation='linear')(inp) # 线性激活是回归的默认选择
    return models.Model(inp, out, name=f"PolynomialRegressor_Degree{degree}")

# 设定多项式次数
degree = 3

# 生成训练数据
x_data = tf.linspace(-20.0, 20.0, 1000) # 从-20到20生成1000个点
y_true = x_data**3 # 目标函数 y = x^3

# 构建多项式特征矩阵 X
# X 的每一行是一个样本的特征向量 [x^0, x^1, x^2, x^3]
X_features = tf.transpose(tf.convert_to_tensor([x_data**p for p in range(degree + 1)], dtype=tf.float32))

# 确保y_true也是float32
y_true = tf.cast(y_true, dtype=tf.float32)

print(f"X_features shape: {X_features.shape}") # 预期 (1000, 4)
print(f"y_true shape: {y_true.shape}")     # 预期 (1000,)

# 实例化模型
model = PolynomialModel(degree)

# 编译模型
model.compile(loss='mean_squared_error', optimizer=optimizers.Adam(learning_rate=0.1))

# 打印模型摘要，查看参数数量
model.summary()

# 训练模型
print("\n开始训练模型...")
history = model.fit(X_features, y_true, epochs=200, verbose=0) # verbose=0 减少输出

# 打印最终损失
print(f"最终训练损失: {history.history['loss'][-1]:.2e}")

# 进行预测
# 预测 x=4 时 y 的值，即 4^3 = 64
test_x_features = tf.constant([[4**0, 4**1, 4**2, 4**3]], dtype=tf.float32)
prediction_4 = model.predict(test_x_features)
print(f"\n预测 4^3 的结果: {prediction_4[0][0]:.2f} (实际值: 64)")

# 预测 x=3 时 y 的值，即 3^3 = 27
test_x_features_3 = tf.constant([[3**0, 3**1, 3**2, 3**3]], dtype=tf.float32)
prediction_3 = model.predict(test_x_features_3)
print(f"预测 3^3 的结果: {prediction_3[0][0]:.2f} (实际值: 27)")

Model: "PolynomialRegressor_Degree3"
_________________________________________________________________
 Layer (type)                Output Shape              Param #   
=================================================================
 input_1 (InputLayer)        [(None, 4)]               0         

 dense (Dense)               (None, 1)                 5         

=================================================================
Total params: 5 (20.00 Byte)
Trainable params: 5 (20.00 Byte)
Non-trainable params: 0 (0.00 Byte)
_________________________________________________________________

开始训练模型...
最终训练损失: 1.44e-11

预测 4^3 的结果: 64.00 (实际值: 64)
预测 3^3 的结果: 27.00 (实际值: 27)