提升神经网络回归模型性能：低x值数据拟合难题求解
2026-7-18
回归拟合模型改进方案
问题背景
我有一组与项目真实数据高度相似的玩具数据：
x = np.array([ 0. , 0.1010101 , 0.2020202 , 0.3030303 , 0.4040404 , 0.50505051, 0.60606061, 0.70707071, 0.80808081, 0.90909091, 1.01010101, 1.11111111, 1.21212121, 1.31313131, 1.41414141, 1.51515152, 1.61616162, 1.71717172, 1.81818182, 1.91919192, 2.02020202, 2.12121212, 2.22222222, 2.32323232, 2.42424242, 2.52525253, 2.62626263, 2.72727273, 2.82828283, 2.92929293, 3.03030303, 3.13131313, 3.23232323, 3.33333333, 3.43434343, 3.53535354, 3.63636364, 3.73737374, 3.83838384, 3.93939394, 4.04040404, 4.14141414, 4.24242424, 4.34343434, 4.44444444, 4.54545455, 4.64646465, 4.74747475, 4.84848485, 4.94949495, 5.05050505, 5.15151515, 5.25252525, 5.35353535, 5.45454545, 5.55555556, 5.65656566, 5.75757576, 5.85858586, 5.95959596, 6.06060606, 6.16161616, 6.26262626, 6.36363636, 6.46464646, 6.56565657, 6.66666667, 6.76767677, 6.86868687, 6.96969697, 7.07070707, 7.17171717, 7.27272727, 7.37373737, 7.47474747, 7.57575758, 7.67676768, 7.77777778, 7.87878788, 7.97979798, 8.08080808, 8.18181818, 8.28282828, 8.38383838, 8.48484848, 8.58585859, 8.68686869, 8.78787879, 8.88888889, 8.98989899, 9.09090909, 9.19191919, 9.29292929, 9.39393939, 9.49494949, 9.5959596 , 9.6969697 , 9.7979798 , 9.8989899 , 10. ]) y = np.array([-0.80373298, 0.76935298, -0.14159923, 1.29519353, 0.3094064 , 0.66238427, 0.42343774, 0.77283061, 1.47505766, 0.45931619, 1.41141125, 1.62579566, 1.28840108, 1.34285815, 0.9329334 , 1.329214 , 1.5139391 , 1.21117778, 0.54639438, 0.51462165, 2.77181805, 1.13110837, 1.86706418, 1.95244603, 1.40661855, 1.30664676, 1.79014375, 1.39412399, 1.17882416, 1.06187797, 1.89504248, 1.50652787, 1.64920352, 2.69228877, 2.24660016, 1.8767469 , 2.22418453, 1.63944449, 1.81288111, 1.59961924, 1.7354012 , 1.65975252, 2.04371439, 2.51920563, 2.3971049 , 1.74297775, 2.22420045, 1.29922847, 1.78963033, 2.76862922, 2.59913081, 2.5868994 , 0.95132831, 2.33654116, 2.14236444, 2.56886641, 2.41801508, 2.03847576, 1.76058536, 1.47914731, 3.22155981, 2.77761667, 2.43482125, 2.87060182, 2.71857598, 2.39742888, 2.55224796, 2.03309053, 2.85056195, 3.01513978, 3.1316874 , 2.14246426, 1.88901478, 2.30135553, 2.90525156, 3.08009528, 2.0941706 , 3.05404934, 3.59780609, 2.32416305, 3.04954219, 1.36782575, 3.16888341, 2.26659839, 2.14637558, 3.26594114, 3.47156645, 3.27828348, 3.48980836, 2.66734284, 2.69708374, 2.90246668, 2.48449401, 3.13271428, 3.08989781, 3.05270477, 3.96243953, 3.28104845, 2.46014121, 3.95762993])
目标是对该数据做回归拟合，得到能复现数据模式的优质模型。试过简单多项式回归、单隐藏层神经网络、双隐藏层神经网络，都没法捕捉x值较低区域的Y轴骤降趋势，求模型改进方法或替代算法。
已尝试的神经网络代码
第一次尝试
from tensorflow import keras import numpy as np import matplotlib.pyplot as plt # Reshape x to be a 2D array of size (N, 1) x = x_data y = y_data x = x.reshape(-1, 1) # Define the model model = keras.models.Sequential([ keras.layers.Dense(10, input_dim=1, activation='relu'), # input layer and hidden layer with 10 neurons keras.layers.Dense(1) # output layer with 1 neuron ]) # Compile the model model.compile(loss='mean_squared_error', optimizer='adam') # Train the model model.fit(x, y, epochs=500, verbose=0) # Make predictions with the model y_pred = model.predict(x) # Plot the original data and the model's predictions plt.scatter(x, y, label='Original data') plt.plot(x, y_pred, color='red', label='Fitted line') plt.legend() plt.show()
第二次尝试
import numpy as np import matplotlib.pyplot as plt from tensorflow import keras x = x_data y = y_data # Preprocess data x = x.reshape(-1,1) # Needs to be reshaped for Keras y = y.reshape(-1,1) # Building the model model = keras.models.Sequential([ keras.layers.Dense(64, activation='relu', input_shape=x.shape[1:]), keras.layers.Dropout(0.2), keras.layers.Dense(64, activation='relu'), keras.layers.Dropout(0.2), keras.layers.Dense(1) ]) # Compile the model model.compile(optimizer='adam', loss='mse') # Train the model history = model.fit(x, y, epochs=50, verbose=0) # Make predictions with the model y_pred = model.predict(x) # Plotting the loss plt.plot(history.history['loss']) plt.title('Model loss') plt.ylabel('Loss') plt.xlabel('Epoch') plt.legend(['Train'], loc='upper right') plt.show() plt.scatter(x, y, label='Original data') plt.plot(x, y_pred, color='red', label='Fitted line') plt.legend() plt.show()
改进方案与替代算法
1. 数据预处理：强化低x区域的权重
低x区域的骤降没被捕捉，核心原因是这部分数据占比小，整体MSE损失对其权重低，模型会优先拟合占比大的高x区域。
- 样本加权：给x<1的样本设置5-10倍权重，让模型更重视这部分数据。Keras中可通过
sample_weight参数实现：# 生成样本权重数组 sample_weights = np.where(x < 1, 5.0, 1.0) # 训练时传入权重 model.fit(x, y, epochs=500, verbose=0, sample_weight=sample_weights.flatten()) - 特征变换：给x添加非线性变换特征（如平方根、对数），降低模型捕捉低x区域趋势的难度：
# 构造多特征输入 x_transformed = np.hstack([x, np.sqrt(x), np.log(x+1)]) # 模型输入维度改为3 model = keras.models.Sequential([ keras.layers.Dense(64, input_dim=3, activation='relu'), keras.layers.Dense(64, activation='relu'), keras.layers.Dense(1) ])
2. 模型结构调整
- 更换激活函数：ReLU在低输入下容易导致神经元“死亡”，改用LeakyReLU或Swish保留低输入下的非线性：
from tensorflow.keras.layers import LeakyReLU model = keras.models.Sequential([ keras.layers.Dense(64, input_dim=1), LeakyReLU(alpha=0.1), keras.layers.Dense(64), LeakyReLU(alpha=0.1), keras.layers.Dense(1) ]) - 分段建模：把x分成低区间（x<1）和高区间（x≥1）分别建模，最后合并结果，让模型针对性拟合局部趋势：
from tensorflow.keras.models import Model from tensorflow.keras.layers import Input, Dense, Concatenate import tensorflow as tf # 输入层 input_layer = Input(shape=(1,)) # 低x分支 low_x_mask = keras.layers.Lambda(lambda x: tf.cast(x < 1, tf.float32))(input_layer) low_x_process = Dense(32, activation='swish')(input_layer) low_x_process = Dense(32, activation='swish')(low_x_process) low_x_out = Dense(1)(low_x_process) # 高x分支 high_x_mask = keras.layers.Lambda(lambda x: tf.cast(x >= 1, tf.float32))(input_layer) high_x_process = Dense(64, activation='swish')(input_layer) high_x_process = Dense(64, activation='swish')(high_x_process) high_x_out = Dense(1)(high_x_process) # 合并分支结果 merged = Concatenate()([low_x_out * low_x_mask, high_x_out * high_x_mask]) output_layer = Dense(1)(merged) # 构建模型 model = Model(inputs=input_layer, outputs=output_layer) model.compile(optimizer
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