一、引言

深度学习的重要性
        深度学习是一种通过模拟人脑神经元结构来进行数据学习和模式识别的技术，在分类任务中展现出强大的能力。

分类任务的多样性

        分类任务涵盖了各种场景，例如簇分类、圆环分类和月牙分类，每种任务都有不同的特征和应用。

二、分类任务详解

2.1、簇分类

• 定义
  簇分类旨在将数据点分为多个簇或类别，目标是在特征空间中找到数据点的天然聚集。
• 数据特性
  通常数据聚集在不同的区域形成簇，这些簇可能具有不同的形状和大小。
• 应用场景
  数据挖掘、市场细分、社交网络分析等。

 簇分类数据

class1_points = np.array(
    [[3.2, 3.0], [2.6, 3.4], [3.5, 4.9], [2.5, 3.4], [1.8, 2.7], [1.3, 1.9], [1.1, 3.4], [1.0, 4.0],
     [1.2, 5.0], [2.8, 4.1],
     [2.7, 3.1], [2.6, 4.5], [2.1, 3.3], [2.3, 2.4], [2.6, 3.1], [1.9, 3.0], [0.7, 4.2], [1.4, 3.3],
     [1.6, 4.6], [2.3, 2.0],
     [1.3, 4.2], [1.9, 3.8], [3.6, 6.0], [1.2, 3.1], [1.6, 3.1], [3.5, 4.1], [1.7, 2.6], [2.4, 3.3],
     [0.8, 2.2], [1.5, 4.3],
     [1.3, 3.9], [1.6, 5.4], [3.4, 3.7], [2.3, 3.4], [2.6, 2.4], [1.8, 2.5], [1.1, 4.1], [1.8, 2.8],
     [0.7, 4.4], [1.1, 3.4],
     [1.9, 3.6], [1.5, 4.9], [1.0, 3.3], [1.4, 3.6], [2.8, 3.3], [3.1, 4.2], [2.7, 3.8], [3.3, 2.6],
     [3.0, 2.7], [0.8, 3.0],
     [1.1, 3.8], [1.8, 3.5], [1.9, 2.8], [0.7, 3.1], [2.5, 2.6], [1.3, 2.5], [2.9, 2.9], [3.1, 2.3],
     [2.4, 2.8], [1.5, 4.0],
     [1.2, 3.8], [2.4, 2.3], [2.1, 1.9], [2.6, 4.2], [2.1, 2.8], [1.6, 2.6], [0.9, 3.8], [1.5, 2.1],
     [1.7, 3.0], [3.0, 2.9],
     [2.3, 2.6], [1.5, 2.9], [2.9, 2.9], [1.9, 2.7], [0.9, 2.7], [1.0, 4.9], [3.3, 4.0], [2.3, 2.7],
     [2.2, 4.0], [1.7, 4.2],
     [1.5, 3.4], [2.1, 3.5], [2.7, 3.9], [1.0, 4.8], [2.4, 2.8], [1.5, 2.6], [2.2, 3.2], [2.5, 2.6],
     [3.9, 2.8], [2.9, 4.1],
     [2.1, 4.3], [1.9, 3.4], [1.3, 1.9], [0.7, 3.3], [1.8, 4.2], [1.7, 3.2], [3.9, 2.9], [1.6, 4.2],
     [2.4, 4.4], [1.8, 1.3],
     [3.5, 2.0], [2.2, 3.1], [3.0, 3.5], [2.9, 3.3], [1.9, 2.9], [1.6, 2.7], [2.8, 3.6], [3.0, 2.7],
     [2.9, 4.4], [3.1, 3.4],
     [1.9, 1.2], [3.0, 1.6], [2.0, 3.7], [1.3, 3.1], [2.8, 2.4], [1.5, 2.6], [2.2, 3.1], [3.0, 3.7],
     [0.9, 4.3], [3.4, 3.6],
     [1.0, 2.4], [2.1, 3.3], [0.7, 2.3], [2.9, 2.3], [2.7, 3.5], [1.3, 2.6], [1.7, 4.2], [2.5, 4.1],
     [2.2, 3.4], [3.3, 3.0],
     [2.2, 3.5], [1.7, 3.1], [1.9, 2.8], [1.7, 2.9], [3.4, 3.0], [1.6, 4.9], [2.8, 3.7], [1.3, 3.7],
     [2.6, 2.6], [4.1, 3.5],
     [4.1, 3.1], [1.2, 2.6], [2.5, 3.0], [1.8, 4.0], [3.6, 4.0], [2.1, 4.3], [1.8, 3.2], [3.3, 1.9],
     [2.4, 3.5], [1.4, 3.9]])
class2_points = np.array(
    [[8.8, 7.2], [7.8, 7.3], [6.8, 7.8], [8.1, 7.5], [7.8, 5.4], [7.6, 8.1], [8.3, 7.5], [6.9, 8.5],
     [8.0, 8.2], [8.7, 7.2],
     [8.8, 7.0], [8.2, 8.3], [7.7, 7.6], [8.3, 8.1], [8.3, 7.7], [8.0, 7.7], [6.7, 6.2], [8.4, 7.8],
     [7.6, 7.3], [6.4, 8.3],
     [8.0, 6.6], [7.0, 6.1], [8.2, 6.5], [6.7, 6.4], [7.1, 8.4], [6.6, 7.6], [7.9, 7.6], [8.0, 8.0],
     [7.3, 8.6], [8.7, 7.5],
     [7.8, 9.2], [7.3, 6.1], [7.7, 7.4], [8.0, 7.3], [8.2, 7.3], [6.5, 8.4], [6.7, 7.0], [7.9, 8.2],
     [6.0, 7.1], [7.9, 7.6],
     [7.1, 7.8], [9.0, 7.4], [7.2, 8.5], [9.1, 6.5], [7.3, 8.6], [7.2, 7.7], [8.8, 7.3], [7.0, 6.5],
     [6.7, 8.4], [7.4, 8.3],
     [9.2, 6.3], [7.8, 8.0], [9.4, 7.3], [8.0, 6.5], [6.8, 7.3], [8.5, 7.4], [6.6, 7.4], [8.6, 8.4],
     [9.8, 6.9], [6.7, 9.5],
     [6.5, 8.0], [8.1, 7.6], [7.4, 8.0], [8.8, 6.1], [7.1, 9.3], [7.3, 7.7], [7.9, 6.7], [7.2, 9.8],
     [8.7, 7.8], [7.8, 9.0],
     [7.2, 7.3], [9.2, 8.9], [7.3, 7.3], [8.3, 6.7], [7.2, 8.2], [8.1, 7.6], [7.5, 9.7], [6.8, 6.9],
     [8.8, 7.5], [7.6, 7.0],
     [7.9, 8.7], [8.8, 7.8], [7.5, 7.0], [8.2, 8.2], [6.9, 6.7], [8.1, 7.8], [8.9, 7.4], [9.4, 7.1],
     [5.8, 7.9], [7.2, 8.0],
     [8.0, 7.2], [7.2, 9.0], [7.3, 7.4], [7.3, 7.9], [9.0, 7.0], [7.9, 7.8], [7.2, 6.9], [8.4, 6.7],
     [8.4, 6.2], [8.4, 7.9],
     [7.6, 6.5], [6.3, 7.0], [8.1, 7.2], [7.2, 7.9], [7.9, 7.0], [7.7, 7.0], [7.1, 7.4], [8.9, 7.7],
     [7.5, 6.3], [7.3, 7.4],
     [8.1, 6.9], [5.4, 8.1], [7.7, 7.1], [7.8, 7.8], [7.3, 8.1], [9.1, 7.5], [7.4, 7.1], [6.6, 7.2],
     [7.7, 7.8], [7.7, 8.8],
     [6.5, 8.4], [8.5, 8.0], [5.9, 8.3], [6.9, 6.4], [7.7, 6.8], [8.5, 6.5], [8.6, 6.5], [8.4, 7.2],
     [8.0, 7.9], [8.3, 8.4],
     [9.2, 7.7], [8.6, 8.0], [7.2, 8.3], [7.6, 8.7], [6.7, 7.5], [6.6, 7.1], [8.7, 8.0], [7.0, 7.8],
     [8.4, 8.9], [6.6, 7.8],
     [8.3, 6.7], [6.7, 7.8], [6.6, 7.1], [8.3, 7.2], [8.9, 8.0], [6.8, 6.6], [8.0, 7.7], [6.3, 7.4],
     [7.2, 8.8], [7.7, 7.4]])

 模型训练效果

2.2、圆环分类

• 定义
  圆环分类任务涉及在特征空间中识别环形结构的数据分布。
• 数据特性
  数据点围绕某个中心形成多个同心圆，每个环对应不同的类别。
• 应用场景
  图像分类、手写数字识别、模式识别等。

圆环分类数据

class1_points = np.array(
    [[1.7, 4.6], [5.4, 7.7], [3.8, 1.9], [3.5, 2.2], [2.2, 2.5], [4.1, 8.1], [3.7, 7.3], [1.8, 4.2],
     [6.8, 2.7], [6.9, 3.1],
     [7.9, 6.9], [8.1, 5.0], [7.2, 7.0], [7.9, 3.8], [6.3, 2.2], [5.0, 2.6], [4.9, 7.6], [6.1, 1.6],
     [3.0, 6.6], [3.3, 6.7],
     [1.8, 4.9], [3.2, 7.5], [7.8, 3.7], [7.3, 2.5], [7.1, 6.7], [1.6, 6.0], [2.6, 2.8], [1.9, 4.3],
     [2.5, 2.8], [7.3, 3.3],
     [7.7, 5.1], [2.7, 7.4], [6.2, 7.7], [5.6, 7.6], [6.4, 7.2], [7.1, 6.6], [3.8, 8.1], [2.4, 6.3],
     [7.5, 3.7], [1.6, 2.9],
     [3.9, 7.8], [7.2, 6.9], [7.4, 4.8], [7.5, 4.4], [2.0, 5.2], [2.0, 4.0], [7.3, 3.8], [5.5, 7.6],
     [7.5, 5.9], [4.0, 2.4],
     [6.9, 7.1], [5.3, 2.0], [3.3, 7.0], [4.0, 2.3], [2.7, 2.7], [5.9, 7.8], [5.7, 2.1], [7.8, 5.9],
     [2.6, 7.0], [5.4, 2.1],
     [7.0, 2.7], [5.4, 7.4], [7.0, 6.4], [7.5, 5.3], [4.2, 2.1], [3.7, 7.7], [7.7, 5.3], [6.1, 7.3],
     [1.6, 4.3], [3.3, 2.4],
     [1.9, 6.4], [1.9, 6.2], [7.7, 6.0], [4.2, 8.4], [4.7, 1.6], [3.0, 3.3], [2.1, 3.6], [1.8, 6.7],
     [4.8, 7.7], [6.8, 2.7],
     [3.3, 2.5], [5.6, 7.5], [5.9, 7.9], [2.3, 4.6], [2.2, 6.2], [4.8, 1.7], [1.9, 4.2], [1.4, 4.1],
     [3.5, 7.1], [5.9, 7.8],
     [6.6, 6.8], [2.3, 5.3], [4.0, 7.6], [3.9, 7.2], [4.6, 2.4], [3.0, 2.2], [7.3, 2.7], [1.6, 5.3],
     [2.8, 2.8], [2.5, 5.7],
     [7.7, 5.6], [4.6, 1.3], [3.1, 7.3], [2.0, 3.1], [7.1, 3.7], [6.1, 7.7], [3.1, 1.9], [6.5, 6.3],
     [2.1, 3.6], [7.3, 5.2],
     [1.7, 6.0], [2.2, 5.0], [7.4, 2.7], [2.2, 6.4], [5.0, 8.2], [2.6, 2.8], [2.6, 2.5], [7.5, 4.0],
     [1.7, 3.7], [3.8, 7.7],
     [2.9, 6.2], [4.9, 1.8], [1.9, 5.3], [6.8, 6.7], [5.2, 1.6], [5.7, 2.3], [3.8, 8.1], [6.7, 3.0],
     [2.3, 3.1], [8.3, 5.8],
     [2.1, 4.5], [5.3, 1.7], [3.2, 1.9], [7.0, 3.1], [6.3, 2.0], [4.2, 7.2], [6.1, 7.4], [2.3, 6.5],
     [5.4, 1.5], [5.7, 7.2],
     [4.5, 7.5], [2.4, 6.8], [7.6, 4.5], [3.3, 2.0], [1.8, 3.6], [1.8, 4.3], [7.5, 4.9], [4.6, 8.3],
     [6.9, 6.8], [7.4, 3.4],
     [3.6, 7.9], [7.6, 4.4], [7.8, 6.1], [6.0, 2.2], [6.4, 2.7], [4.9, 7.6], [1.7, 6.4], [7.7, 5.7],
     [6.8, 6.8], [3.1, 2.9],
     [2.0, 2.5], [4.5, 2.3], [6.7, 7.2], [7.5, 7.1], [1.9, 5.5], [5.5, 1.7], [6.6, 2.2], [6.1, 7.2],
     [3.9, 2.1], [2.5, 6.6],
     [7.7, 3.9], [7.4, 5.5], [7.6, 3.8], [3.7, 2.2], [2.3, 7.3], [5.0, 2.2], [5.5, 1.4], [2.9, 7.0],
     [6.7, 2.4], [2.0, 5.6],
     [6.4, 2.6], [7.3, 4.9], [4.0, 1.6], [3.3, 2.3], [7.6, 5.1], [3.5, 1.5], [4.7, 7.9], [6.1, 7.4],
     [2.2, 6.2], [6.9, 2.6],
     [2.2, 2.7], [4.1, 7.5], [8.2, 4.4], [3.5, 7.8], [2.4, 6.5], [2.1, 3.8], [1.8, 5.1], [2.3, 2.6],
     [6.4, 2.7], [7.0, 2.6],
     [7.4, 3.6], [5.9, 1.7], [8.3, 5.8], [7.8, 3.6], [7.7, 5.1], [8.0, 3.9], [1.3, 5.3], [3.4, 7.1],
     [4.7, 7.8], [2.1, 3.8],
     [7.1, 6.0], [7.5, 4.1], [7.1, 3.5], [7.3, 6.9], [6.6, 2.3], [7.5, 3.3], [7.1, 6.5], [8.0, 5.8],
     [8.0, 4.2], [3.6, 7.7],
     [1.9, 5.0], [2.6, 2.8], [5.1, 7.0], [6.9, 7.2], [2.0, 6.0], [7.5, 2.5], [4.0, 2.1], [2.9, 7.0],
     [4.2, 7.2], [5.3, 1.8],
     [2.6, 6.8], [3.1, 2.3], [3.6, 2.3], [5.5, 1.3], [1.3, 4.2], [6.2, 1.9], [2.5, 3.1], [1.8, 4.5],
     [1.7, 5.5], [5.7, 7.8],
     [8.2, 4.8], [2.0, 3.4], [1.4, 4.4], [5.5, 7.9], [4.0, 1.7], [7.8, 4.7], [6.3, 7.2], [2.5, 2.3],
     [7.4, 4.4], [5.1, 7.9]])
class2_points = np.array(
    [[5.7, 4.8], [4.8, 5.0], [4.7, 4.6], [4.6, 5.3], [5.0, 5.5], [4.3, 4.9], [4.2, 5.9], [6.0, 5.0],
     [4.1, 5.2], [5.4, 5.0],
     [4.9, 5.4], [4.5, 6.2], [5.3, 5.5], [4.2, 5.0], [4.0, 4.9], [5.9, 4.9], [4.3, 6.1], [4.5, 4.3],
     [5.1, 5.8], [5.6, 4.5],
     [4.9, 4.3], [5.5, 5.7], [5.4, 5.0], [4.7, 4.9], [5.6, 5.3], [5.8, 4.8], [4.8, 5.6], [5.3, 5.3],
     [5.1, 4.7], [5.0, 5.3],
     [4.0, 4.4], [5.9, 5.2], [5.7, 4.7], [5.8, 5.2], [5.1, 4.0], [5.8, 5.9], [5.3, 6.0], [5.5, 4.8],
     [5.1, 4.7], [4.7, 4.3],
     [5.7, 5.0], [4.3, 4.7], [5.7, 4.9], [4.7, 4.0], [4.9, 4.9], [5.2, 4.6], [4.6, 5.6], [5.2, 5.3],
     [4.8, 5.9], [4.5, 4.7],
     [5.3, 5.2], [4.7, 4.3], [4.7, 5.7], [4.7, 4.2], [4.7, 5.3], [5.3, 5.4], [5.4, 5.9], [4.6, 4.1],
     [4.1, 5.8], [5.6, 5.1],
     [5.2, 4.5], [5.6, 4.7], [5.0, 4.8], [5.7, 4.3], [4.5, 5.7], [4.4, 5.7], [5.5, 5.3], [4.7, 5.4],
     [5.1, 5.7], [5.2, 4.3],
     [4.6, 4.9], [4.7, 5.5], [4.5, 4.2], [5.2, 4.5], [5.4, 3.9], [4.0, 5.0], [4.4, 4.0], [5.0, 4.2],
     [5.8, 5.6], [5.8, 5.2],
     [4.7, 4.6], [4.7, 5.8], [5.6, 4.5], [5.8, 4.9], [4.6, 5.5], [5.6, 4.5], [5.1, 4.5], [4.2, 4.8],
     [4.9, 5.3], [5.0, 5.2],
     [4.0, 4.8], [5.5, 4.8], [6.0, 4.7], [4.4, 5.1], [4.3, 4.9], [5.1, 5.6], [4.7, 5.6], [5.1, 4.9],
     [4.2, 5.4], [4.4, 4.6],
     [5.5, 5.9], [4.1, 4.8], [5.0, 4.6], [5.2, 5.0], [4.1, 5.5], [4.6, 5.1], [5.2, 5.5], [5.1, 4.0],
     [4.4, 4.5], [5.3, 5.3],
     [4.8, 5.3], [5.2, 4.6], [5.7, 4.4], [4.3, 5.0], [5.1, 4.9], [4.6, 5.0], [5.4, 5.6], [5.3, 4.4],
     [4.6, 4.3], [5.2, 5.6],
     [5.0, 4.3], [4.4, 4.4], [5.5, 4.9], [4.3, 5.5], [5.0, 5.3], [4.8, 4.9], [5.3, 5.6], [4.1, 4.7],
     [4.6, 5.2], [5.5, 4.6],
     [4.6, 4.6], [4.5, 5.4], [4.6, 4.2], [5.1, 4.3], [5.2, 4.3], [5.1, 5.6], [5.5, 4.5], [5.1, 4.0],
     [4.5, 5.1], [4.8, 3.7],
     [4.3, 5.1], [4.6, 5.4], [5.2, 3.9], [4.6, 5.1], [4.2, 5.1], [4.5, 5.2], [5.6, 5.3], [5.6, 5.1],
     [5.9, 5.2], [5.0, 4.1],
     [5.1, 4.3], [4.8, 6.0], [5.3, 5.5], [5.3, 4.4], [4.4, 5.1], [5.2, 5.0], [4.9, 4.4], [5.3, 5.2],
     [5.2, 6.1], [5.6, 5.9],
     [4.7, 4.2], [6.1, 5.6], [4.6, 5.7], [5.5, 5.0], [4.5, 5.1], [4.8, 6.0], [4.8, 5.0], [5.5, 4.3],
     [4.1, 4.9], [3.9, 4.6],
     [4.9, 5.3], [4.4, 4.1], [4.6, 5.3], [5.0, 4.7], [5.3, 5.9], [5.1, 5.4], [5.3, 5.3], [4.9, 4.5],
     [5.6, 5.1], [5.2, 4.5],
     [5.3, 4.6], [5.5, 5.6], [5.0, 6.1], [4.5, 5.3], [4.8, 5.6], [4.7, 4.9], [4.7, 5.6], [4.6, 4.3],
     [5.8, 5.0], [4.9, 4.8],
     [5.6, 5.3], [5.5, 5.2], [4.8, 5.3], [4.6, 4.5], [5.2, 4.9], [5.5, 5.6], [6.2, 4.1], [5.6, 5.3],
     [5.3, 5.4], [5.4, 5.0],
     [5.5, 4.8], [5.1, 4.6], [4.8, 5.4], [4.8, 5.3], [5.8, 4.8], [4.5, 4.8], [4.6, 4.9], [4.3, 3.9],
     [4.6, 5.3], [5.1, 5.3],
     [5.4, 5.7], [4.3, 5.2], [4.8, 4.9], [5.6, 4.7], [4.2, 5.0], [5.3, 5.6], [4.9, 4.0], [5.1, 4.7],
     [5.0, 5.4], [6.0, 5.5],
     [5.5, 4.6], [5.7, 5.3], [4.5, 4.7], [5.5, 5.0], [5.9, 4.9], [5.5, 4.6], [4.9, 5.6], [5.4, 5.3],
     [5.2, 4.4], [4.3, 4.5],
     [5.1, 4.2], [4.3, 5.1], [5.6, 5.7], [4.8, 5.0], [5.1, 5.5], [5.7, 5.2], [5.9, 4.9], [5.1, 4.3],
     [5.3, 5.2], [4.4, 4.7],
     [5.2, 5.8], [6.3, 5.1], [4.0, 5.4], [5.4, 4.7], [4.2, 5.3], [5.7, 4.9], [5.4, 5.5], [4.8, 5.2],
     [5.4, 5.8], [4.6, 5.0]])

 模型训练效果 

2.3、月牙分类

• 定义
  月牙分类任务要求识别流形或不规则的形状，数据分布呈现出像月牙形状的特征。
• 数据特性
  数据集中的点通常呈现出一种弯曲的形态，具有独特的边界。
• 应用场景
  生物医学影像分析、信号处理、推荐系统等。

 月牙分类数据

class1_points = np.array(
    [[6.5, 4.3], [4.5, 6.4], [1.3, 5.1], [1.7, 4.4], [4.8, 5.7], [5.4, 5.6], [1.8, 4.9], [1.2, 3.8],
     [2.8, 5.7], [6.4, 3.8],
     [4.5, 5.9], [5.3, 6.0], [5.9, 5.0], [1.7, 4.6], [2.3, 5.7], [3.4, 6.1], [5.9, 4.4], [5.4, 5.1],
     [5.2, 5.2], [5.6, 5.4],
     [4.2, 6.2], [1.4, 3.7], [3.6, 6.3], [4.8, 6.0], [4.8, 6.0], [5.0, 6.1], [5.8, 5.1], [1.6, 4.5],
     [1.5, 5.1], [2.2, 6.0],
     [5.1, 5.8], [3.8, 6.3], [2.0, 5.7], [2.1, 5.6], [2.0, 5.1], [1.0, 4.9], [3.0, 6.3], [6.0, 4.2],
     [2.3, 6.3], [4.8, 6.1],
     [1.8, 5.1], [2.2, 5.7], [6.3, 4.3], [5.7, 5.3], [5.6, 5.5], [3.0, 6.1], [6.1, 3.7], [6.3, 4.7],
     [3.4, 6.1], [5.2, 5.7],
     [5.8, 3.7], [0.7, 4.6], [4.9, 6.2], [1.8, 5.1], [4.6, 5.9], [1.5, 5.0], [1.4, 4.4], [4.0, 6.4],
     [5.3, 5.8], [4.6, 6.1],
     [3.5, 6.0], [6.2, 4.6], [4.5, 6.0], [2.6, 6.1], [5.9, 5.0], [2.8, 6.4], [2.4, 6.0], [5.3, 6.0],
     [2.0, 5.7], [1.2, 3.7],
     [2.8, 5.9], [2.5, 5.5], [6.3, 4.6], [1.2, 3.7], [6.3, 4.4], [6.0, 4.8], [1.5, 4.2], [6.4, 4.2],
     [1.3, 4.6], [2.0, 5.2],
     [1.9, 5.2], [1.6, 5.4], [5.5, 5.7], [3.5, 6.6], [1.7, 5.0], [6.2, 4.6], [6.1, 4.5], [4.1, 5.9],
     [6.1, 4.9], [1.7, 5.2],
     [3.5, 6.2], [2.9, 6.4], [5.0, 5.8], [2.5, 5.8], [3.1, 6.0], [2.0, 5.1], [2.6, 5.7], [6.1, 4.0],
     [6.5, 4.4], [5.4, 6.1],
     [5.9, 4.1], [4.7, 5.9], [2.4, 6.5], [4.5, 6.4], [5.9, 4.6], [0.9, 3.9], [3.6, 6.3], [3.7, 6.3],
     [1.6, 4.3], [6.0, 5.7],
     [4.2, 6.3], [1.8, 5.2], [2.7, 5.9], [2.4, 5.5], [6.4, 3.8], [5.2, 6.1], [6.2, 4.7], [4.2, 6.5],
     [5.7, 3.6], [3.9, 6.1],
     [1.1, 4.6], [5.5, 5.3], [2.0, 5.9], [5.2, 5.4], [5.7, 5.2], [5.3, 5.0], [1.4, 4.1], [2.8, 6.6],
     [3.6, 6.3], [1.1, 4.3],
     [5.5, 5.2], [3.9, 6.9], [6.2, 4.2], [5.5, 5.5], [1.6, 4.1], [1.1, 3.9], [1.4, 4.9], [4.5, 6.1],
     [1.7, 5.0], [1.9, 4.7],
     [5.8, 5.7], [4.8, 5.6], [3.2, 5.7], [6.3, 4.0], [1.6, 4.2], [1.8, 5.1], [1.9, 5.5], [2.9, 5.6],
     [1.0, 3.8], [5.9, 5.5],
     [2.6, 5.6], [5.3, 5.4], [1.5, 5.0], [3.2, 6.1], [1.0, 4.1], [1.9, 5.8], [3.3, 6.2], [6.1, 3.9],
     [2.9, 5.8], [4.8, 5.9],
     [6.0, 4.4], [3.6, 6.2], [1.6, 5.1], [5.6, 5.0], [4.0, 6.2], [6.2, 4.3], [4.2, 6.4], [4.0, 6.1],
     [5.5, 5.1], [4.3, 6.1],
     [4.5, 5.8], [3.7, 6.7], [1.6, 5.6], [5.7, 4.6], [1.6, 4.9], [6.2, 5.7], [2.8, 6.2], [2.1, 5.7],
     [5.8, 6.2], [1.5, 5.0],
     [5.6, 5.6], [4.1, 5.7], [1.8, 4.6], [6.4, 4.1], [1.2, 3.8], [2.4, 6.0], [1.5, 5.2], [6.0, 3.9],
     [5.9, 4.7], [1.9, 5.5],
     [2.3, 5.5], [6.1, 4.4], [2.0, 5.2], [1.8, 5.5], [4.6, 6.3], [3.4, 6.2], [4.7, 6.3], [3.1, 6.1],
     [3.8, 6.3], [5.7, 5.5],
     [1.9, 5.4], [4.7, 5.9], [6.0, 4.2], [4.5, 6.5], [1.3, 4.2], [5.1, 6.0], [1.8, 5.2], [4.0, 6.4],
     [5.8, 5.6], [1.2, 3.9],
     [6.1, 5.4], [1.7, 4.9], [6.3, 5.0], [5.2, 5.0], [3.0, 6.4], [1.6, 4.8], [1.5, 5.2], [4.7, 6.3],
     [1.5, 4.8], [5.3, 5.8],
     [4.3, 5.9], [3.2, 6.3], [2.4, 5.5], [2.6, 5.4], [1.2, 3.9], [4.8, 6.3], [6.2, 4.6], [1.3, 5.3],
     [6.6, 4.1], [2.9, 6.3],
     [3.3, 6.1], [6.0, 5.3], [1.5, 4.9], [5.6, 5.7], [5.9, 4.5], [4.9, 6.1], [6.0, 4.6], [5.0, 5.4],
     [3.4, 6.1], [5.9, 4.9],
     [2.8, 5.4], [1.9, 5.3], [3.2, 5.8], [1.2, 4.7], [3.1, 6.3], [1.2, 4.0], [6.0, 5.7], [2.7, 6.0],
     [3.4, 6.0], [5.9, 5.4]])
class2_points = np.array(
    [[6.5, 2.5], [6.4, 2.3], [6.6, 2.8], [7.0, 2.6], [4.3, 2.9], [4.1, 3.7], [3.9, 3.3], [7.2, 2.7],
     [3.8, 4.5], [4.0, 4.7],
     [4.0, 3.9], [8.3, 3.8], [6.5, 3.1], [8.0, 3.6], [7.9, 3.4], [6.8, 2.5], [4.0, 4.4], [7.0, 2.6],
     [7.7, 3.1], [6.0, 2.1],
     [6.7, 2.7], [8.7, 4.2], [4.0, 3.9], [5.9, 2.2], [6.3, 2.7], [7.3, 2.9], [5.0, 2.6], [8.1, 3.9],
     [4.2, 4.0], [5.1, 2.5],
     [8.2, 3.3], [7.1, 2.9], [5.0, 3.0], [7.1, 2.3], [4.8, 3.1], [3.5, 4.4], [8.3, 3.3], [5.2, 3.0],
     [6.1, 2.2], [6.8, 2.2],
     [3.9, 4.9], [8.6, 3.6], [6.0, 2.3], [4.1, 4.0], [5.2, 2.8], [8.2, 3.5], [8.1, 3.4], [8.7, 4.9],
     [5.0, 2.4], [5.0, 2.6],
     [8.0, 3.0], [8.4, 4.3], [5.3, 2.7], [8.7, 5.1], [5.6, 2.5], [5.4, 2.7], [3.8, 4.5], [9.1, 4.3],
     [8.8, 4.1], [4.7, 3.3],
     [8.4, 4.6], [8.3, 4.5], [7.0, 2.7], [6.4, 2.3], [5.2, 2.5], [7.0, 2.2], [8.6, 3.3], [7.5, 3.0],
     [4.0, 3.9], [7.6, 3.0],
     [7.0, 2.7], [4.3, 3.1], [5.7, 2.8], [3.8, 4.3], [4.9, 3.1], [4.1, 3.3], [7.0, 2.3], [5.1, 2.9],
     [8.9, 4.5], [6.0, 2.7],
     [7.4, 2.6], [8.7, 4.7], [8.6, 4.5], [7.7, 3.0], [8.9, 5.0], [4.1, 4.0], [3.9, 4.8], [3.7, 3.8],
     [5.5, 2.3], [7.5, 3.4],
     [4.2, 3.3], [4.1, 3.5], [7.8, 3.1], [3.8, 4.7], [5.2, 3.3], [3.5, 4.7], [3.5, 4.8], [3.9, 4.2],
     [6.7, 3.1], [7.9, 3.0],
     [8.6, 4.1], [8.5, 4.4], [7.3, 2.6], [3.4, 4.7], [8.7, 3.9], [7.6, 3.0], [4.6, 3.1], [4.8, 2.7],
     [4.5, 2.5], [7.4, 2.9],
     [5.1, 2.7], [6.9, 2.7], [7.6, 2.6], [9.0, 5.0], [7.1, 2.2], [5.0, 2.7], [5.6, 2.4], [3.6, 4.8],
     [6.0, 2.4], [6.9, 2.9],
     [8.3, 4.9], [3.9, 4.0], [4.9, 3.1], [8.7, 3.9], [6.3, 2.4], [6.8, 2.5], [5.8, 2.1], [4.5, 4.1],
     [4.7, 3.2], [6.3, 2.6],
     [8.8, 4.8], [8.6, 4.1], [4.5, 3.8], [3.6, 4.3], [8.8, 5.0], [4.2, 3.9], [8.6, 4.4], [8.8, 4.0],
     [5.0, 3.4], [6.4, 2.5],
     [4.6, 2.6], [6.0, 2.6], [8.1, 3.5], [8.7, 4.5], [4.8, 2.8], [5.9, 2.7], [6.8, 2.6], [8.9, 4.6],
     [6.4, 2.6], [6.9, 2.5],
     [8.8, 3.3], [3.7, 4.0], [8.3, 4.0], [3.6, 4.3], [7.2, 2.2], [8.8, 4.4], [8.7, 4.7], [3.8, 4.4],
     [8.1, 3.4], [3.5, 4.7],
     [8.7, 4.1], [4.3, 3.8], [3.6, 4.0], [5.0, 2.7], [7.7, 3.2], [8.4, 3.2], [4.3, 3.7], [8.6, 4.3],
     [7.5, 3.2], [8.3, 3.8],
     [4.9, 2.9], [5.4, 2.4], [3.9, 4.9], [8.9, 3.6], [8.3, 3.4], [8.2, 3.3], [7.8, 2.8], [8.2, 3.2],
     [8.9, 4.8], [8.6, 3.8],
     [3.9, 5.3], [4.4, 4.6], [7.8, 3.0], [6.9, 2.7], [7.7, 3.0], [3.7, 3.7], [6.6, 3.0], [5.3, 2.6],
     [4.4, 4.1], [8.1, 3.6],
     [8.5, 3.4], [8.0, 3.7], [5.2, 2.7], [7.3, 2.8], [4.1, 4.0], [8.5, 3.6], [7.5, 2.4], [3.9, 3.8],
     [5.9, 2.5], [6.6, 2.9],
     [4.4, 3.4], [4.8, 3.3], [4.4, 3.1], [8.7, 4.8], [6.2, 2.7], [5.0, 3.2], [5.6, 2.7], [8.5, 4.2],
     [4.2, 3.5], [4.0, 3.1],
     [3.8, 4.1], [5.3, 2.2], [4.9, 3.3], [5.7, 3.1], [4.4, 3.5], [5.3, 2.8], [4.2, 3.3], [8.4, 3.6],
     [8.1, 3.5], [3.8, 4.4],
     [3.6, 4.3], [4.3, 4.6], [7.9, 3.1], [8.9, 4.9], [7.8, 3.2], [4.1, 3.7], [4.8, 3.1], [3.7, 4.3],
     [8.5, 3.8], [5.2, 2.7],
     [7.3, 2.8], [6.5, 2.6], [8.4, 4.3], [8.2, 4.0], [7.2, 2.9], [3.7, 4.2], [7.6, 2.6], [4.3, 4.7],
     [4.5, 3.5], [4.0, 4.2],
     [6.4, 2.7], [6.3, 2.6], [8.9, 3.9], [5.8, 2.3], [6.1, 2.6], [4.1, 3.7], [8.2, 3.1], [9.1, 4.5],
     [3.7, 4.1], [6.3, 2.7]])

  模型训练效果 

三、PyTorch实现

 以月牙分类为例

 划分数据集

# 将 point1 分割为训练集和测试集
np.random.shuffle(class1_points)  # 随机打乱数据
split_index = int(0.1 * len(class1_points))  # 取前 10% 的数据作为测试集

class1_train_points = class1_points[split_index:]
class2_train_points = class2_points[split_index:]
class1_test_points = class1_points[:split_index]
class2_test_points = class2_points[:split_index]

# 合并两类点
train_points = np.concatenate((class1_train_points, class2_train_points))
# 标签 0表示类别1，1表示类别2
train_labels1 = np.zeros(len(class1_train_points))
train_labels2 = np.ones(len(class2_train_points))
train_labels = np.concatenate((train_labels1, train_labels2))
# 合并两类点
test_points = np.concatenate((class1_test_points, class2_test_points))
# 标签 0表示类别1，1表示类别2
test_labels1 = np.zeros(len(class1_test_points))
test_labels2 = np.ones(len(class2_test_points))
test_labels = np.concatenate((test_labels1, test_labels2))

 构建模型

class ModelClass(nn.Module):
    def __init__(self):
        super().__init__()
        self.layer1 = nn.Linear(2, 8)
        self.layer2 = nn.Linear(8, 16)
        self.layer3 = nn.Linear(16, 32)
        self.layer4 = nn.Linear(32, 16)
        self.layer5 = nn.Linear(16, 8)
        self.layer6 = nn.Linear(8, 2)

    def forward(self, x):
        x = torch.tanh(self.layer1(x))
        x = torch.tanh(self.layer2(x))
        x = torch.tanh(self.layer3(x))
        x = torch.tanh(self.layer4(x))
        x = torch.tanh(self.layer5(x))
        x = torch.softmax(self.layer6(x),dim=1)
        return x


model = ModelClass()

 创建损失函数和优化器

criterion = nn.CrossEntropyLoss()
optimizer = optim.Adam(model.parameters(), lr=0.01, weight_decay=0.005)

 模型训练

for n in range(1,2001):
    # 将numpy数据转换为torch tensor
    inputs = torch.tensor(train_points, dtype=torch.float32)
    train_labels = torch.tensor(train_labels, dtype=torch.long)

    # 前向传播
    outputs = model(inputs)
    loss = criterion(outputs, train_labels)

    # 反向传播和优化
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()
    if n % 100== 0 or n == 1:
        print(n,loss.item())

 可视化

# 创建等高线绘图的网格点
x_min, x_max = 0, 10
y_min, y_max = 0, 10
step_size = 0.2
xx, yy = np.meshgrid(np.arange(x_min, x_max, step_size),
                     np.arange(y_min, y_max, step_size))
grid_points = np.c_[xx.ravel(), yy.ravel()]

# 创建三维图形和右侧的二维子图
fig = plt.figure(figsize=(10, 5))

ax1 = fig.add_subplot(121)
ax2 = fig.add_subplot(122)

step_list = []
loss_list = []
test_step_list = []
test_loss_list = []

# 开始迭代
for n in range(1,2001):
    # 将numpy数据转换为torch tensor
    inputs = torch.tensor(train_points, dtype=torch.float32)
    train_labels = torch.tensor(train_labels, dtype=torch.long)

    # 前向传播
    outputs = model(inputs)
    loss = criterion(outputs, train_labels)

    # 反向传播和优化
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

    # 更新右侧的损失图数据并绘制
    step_list.append(n)
    loss_list.append(loss.detach())

    # 显示频率设置
    frequency_display = 50
    # 显示与输出
    if n % 100== 0 or n == 1:
        # 使用训练好的模型预测网格点的标签
        grid_points_tensor = torch.tensor(grid_points, dtype=torch.float32)
        Z = model(grid_points_tensor).detach().numpy()
        Z = Z[:, 1]  # 取正类的概率值
        Z = Z.reshape(xx.shape)

        # 绘制2D图
        ax1 = plt.subplot(121)
        ax1.clear()
        ax1.scatter(class1_train_points[:, 0], class1_train_points[:, 1], c='blue', label='label1')
        ax1.scatter(class2_train_points[:, 0], class2_train_points[:, 1], c='red', label='label2')
        ax1.contour(xx, yy, Z, levels=[0.5], colors='black')

        # 计算测试集损失
        test_inputs = torch.tensor(test_points, dtype=torch.float32)
        y_pred_test = model(test_inputs)
        test_labels = torch.tensor(test_labels, dtype=torch.long)
        loss_test = criterion(y_pred_test, test_labels)
        test_step_list.append(n)
        test_loss_list.append(loss_test.detach())

        ax2 = plt.subplot(122)
        ax2.clear()
        ax2.plot(step_list, loss_list, 'r-', label='Train Loss')
        ax2.plot(test_step_list, test_loss_list, 'b-', label='Test Loss')  # 绘制测试集损失
        ax2.set_xlabel("Step")
        ax2.set_ylabel("Loss")
        ax2.legend()

plt.show()

 完整代码

import numpy as np
import torch
import random
import torch.nn as nn
import torch.optim as optim
import matplotlib.pyplot as plt
import torch.nn.init as init

# 创造数据，数据集
class1_points = np.array(
    [[6.5, 4.3], [4.5, 6.4], [1.3, 5.1], [1.7, 4.4], [4.8, 5.7], [5.4, 5.6], [1.8, 4.9], [1.2, 3.8],
     [2.8, 5.7], [6.4, 3.8],
     [4.5, 5.9], [5.3, 6.0], [5.9, 5.0], [1.7, 4.6], [2.3, 5.7], [3.4, 6.1], [5.9, 4.4], [5.4, 5.1],
     [5.2, 5.2], [5.6, 5.4],
     [4.2, 6.2], [1.4, 3.7], [3.6, 6.3], [4.8, 6.0], [4.8, 6.0], [5.0, 6.1], [5.8, 5.1], [1.6, 4.5],
     [1.5, 5.1], [2.2, 6.0],
     [5.1, 5.8], [3.8, 6.3], [2.0, 5.7], [2.1, 5.6], [2.0, 5.1], [1.0, 4.9], [3.0, 6.3], [6.0, 4.2],
     [2.3, 6.3], [4.8, 6.1],
     [1.8, 5.1], [2.2, 5.7], [6.3, 4.3], [5.7, 5.3], [5.6, 5.5], [3.0, 6.1], [6.1, 3.7], [6.3, 4.7],
     [3.4, 6.1], [5.2, 5.7],
     [5.8, 3.7], [0.7, 4.6], [4.9, 6.2], [1.8, 5.1], [4.6, 5.9], [1.5, 5.0], [1.4, 4.4], [4.0, 6.4],
     [5.3, 5.8], [4.6, 6.1],
     [3.5, 6.0], [6.2, 4.6], [4.5, 6.0], [2.6, 6.1], [5.9, 5.0], [2.8, 6.4], [2.4, 6.0], [5.3, 6.0],
     [2.0, 5.7], [1.2, 3.7],
     [2.8, 5.9], [2.5, 5.5], [6.3, 4.6], [1.2, 3.7], [6.3, 4.4], [6.0, 4.8], [1.5, 4.2], [6.4, 4.2],
     [1.3, 4.6], [2.0, 5.2],
     [1.9, 5.2], [1.6, 5.4], [5.5, 5.7], [3.5, 6.6], [1.7, 5.0], [6.2, 4.6], [6.1, 4.5], [4.1, 5.9],
     [6.1, 4.9], [1.7, 5.2],
     [3.5, 6.2], [2.9, 6.4], [5.0, 5.8], [2.5, 5.8], [3.1, 6.0], [2.0, 5.1], [2.6, 5.7], [6.1, 4.0],
     [6.5, 4.4], [5.4, 6.1],
     [5.9, 4.1], [4.7, 5.9], [2.4, 6.5], [4.5, 6.4], [5.9, 4.6], [0.9, 3.9], [3.6, 6.3], [3.7, 6.3],
     [1.6, 4.3], [6.0, 5.7],
     [4.2, 6.3], [1.8, 5.2], [2.7, 5.9], [2.4, 5.5], [6.4, 3.8], [5.2, 6.1], [6.2, 4.7], [4.2, 6.5],
     [5.7, 3.6], [3.9, 6.1],
     [1.1, 4.6], [5.5, 5.3], [2.0, 5.9], [5.2, 5.4], [5.7, 5.2], [5.3, 5.0], [1.4, 4.1], [2.8, 6.6],
     [3.6, 6.3], [1.1, 4.3],
     [5.5, 5.2], [3.9, 6.9], [6.2, 4.2], [5.5, 5.5], [1.6, 4.1], [1.1, 3.9], [1.4, 4.9], [4.5, 6.1],
     [1.7, 5.0], [1.9, 4.7],
     [5.8, 5.7], [4.8, 5.6], [3.2, 5.7], [6.3, 4.0], [1.6, 4.2], [1.8, 5.1], [1.9, 5.5], [2.9, 5.6],
     [1.0, 3.8], [5.9, 5.5],
     [2.6, 5.6], [5.3, 5.4], [1.5, 5.0], [3.2, 6.1], [1.0, 4.1], [1.9, 5.8], [3.3, 6.2], [6.1, 3.9],
     [2.9, 5.8], [4.8, 5.9],
     [6.0, 4.4], [3.6, 6.2], [1.6, 5.1], [5.6, 5.0], [4.0, 6.2], [6.2, 4.3], [4.2, 6.4], [4.0, 6.1],
     [5.5, 5.1], [4.3, 6.1],
     [4.5, 5.8], [3.7, 6.7], [1.6, 5.6], [5.7, 4.6], [1.6, 4.9], [6.2, 5.7], [2.8, 6.2], [2.1, 5.7],
     [5.8, 6.2], [1.5, 5.0],
     [5.6, 5.6], [4.1, 5.7], [1.8, 4.6], [6.4, 4.1], [1.2, 3.8], [2.4, 6.0], [1.5, 5.2], [6.0, 3.9],
     [5.9, 4.7], [1.9, 5.5],
     [2.3, 5.5], [6.1, 4.4], [2.0, 5.2], [1.8, 5.5], [4.6, 6.3], [3.4, 6.2], [4.7, 6.3], [3.1, 6.1],
     [3.8, 6.3], [5.7, 5.5],
     [1.9, 5.4], [4.7, 5.9], [6.0, 4.2], [4.5, 6.5], [1.3, 4.2], [5.1, 6.0], [1.8, 5.2], [4.0, 6.4],
     [5.8, 5.6], [1.2, 3.9],
     [6.1, 5.4], [1.7, 4.9], [6.3, 5.0], [5.2, 5.0], [3.0, 6.4], [1.6, 4.8], [1.5, 5.2], [4.7, 6.3],
     [1.5, 4.8], [5.3, 5.8],
     [4.3, 5.9], [3.2, 6.3], [2.4, 5.5], [2.6, 5.4], [1.2, 3.9], [4.8, 6.3], [6.2, 4.6], [1.3, 5.3],
     [6.6, 4.1], [2.9, 6.3],
     [3.3, 6.1], [6.0, 5.3], [1.5, 4.9], [5.6, 5.7], [5.9, 4.5], [4.9, 6.1], [6.0, 4.6], [5.0, 5.4],
     [3.4, 6.1], [5.9, 4.9],
     [2.8, 5.4], [1.9, 5.3], [3.2, 5.8], [1.2, 4.7], [3.1, 6.3], [1.2, 4.0], [6.0, 5.7], [2.7, 6.0],
     [3.4, 6.0], [5.9, 5.4]])
class2_points = np.array(
    [[6.5, 2.5], [6.4, 2.3], [6.6, 2.8], [7.0, 2.6], [4.3, 2.9], [4.1, 3.7], [3.9, 3.3], [7.2, 2.7],
     [3.8, 4.5], [4.0, 4.7],
     [4.0, 3.9], [8.3, 3.8], [6.5, 3.1], [8.0, 3.6], [7.9, 3.4], [6.8, 2.5], [4.0, 4.4], [7.0, 2.6],
     [7.7, 3.1], [6.0, 2.1],
     [6.7, 2.7], [8.7, 4.2], [4.0, 3.9], [5.9, 2.2], [6.3, 2.7], [7.3, 2.9], [5.0, 2.6], [8.1, 3.9],
     [4.2, 4.0], [5.1, 2.5],
     [8.2, 3.3], [7.1, 2.9], [5.0, 3.0], [7.1, 2.3], [4.8, 3.1], [3.5, 4.4], [8.3, 3.3], [5.2, 3.0],
     [6.1, 2.2], [6.8, 2.2],
     [3.9, 4.9], [8.6, 3.6], [6.0, 2.3], [4.1, 4.0], [5.2, 2.8], [8.2, 3.5], [8.1, 3.4], [8.7, 4.9],
     [5.0, 2.4], [5.0, 2.6],
     [8.0, 3.0], [8.4, 4.3], [5.3, 2.7], [8.7, 5.1], [5.6, 2.5], [5.4, 2.7], [3.8, 4.5], [9.1, 4.3],
     [8.8, 4.1], [4.7, 3.3],
     [8.4, 4.6], [8.3, 4.5], [7.0, 2.7], [6.4, 2.3], [5.2, 2.5], [7.0, 2.2], [8.6, 3.3], [7.5, 3.0],
     [4.0, 3.9], [7.6, 3.0],
     [7.0, 2.7], [4.3, 3.1], [5.7, 2.8], [3.8, 4.3], [4.9, 3.1], [4.1, 3.3], [7.0, 2.3], [5.1, 2.9],
     [8.9, 4.5], [6.0, 2.7],
     [7.4, 2.6], [8.7, 4.7], [8.6, 4.5], [7.7, 3.0], [8.9, 5.0], [4.1, 4.0], [3.9, 4.8], [3.7, 3.8],
     [5.5, 2.3], [7.5, 3.4],
     [4.2, 3.3], [4.1, 3.5], [7.8, 3.1], [3.8, 4.7], [5.2, 3.3], [3.5, 4.7], [3.5, 4.8], [3.9, 4.2],
     [6.7, 3.1], [7.9, 3.0],
     [8.6, 4.1], [8.5, 4.4], [7.3, 2.6], [3.4, 4.7], [8.7, 3.9], [7.6, 3.0], [4.6, 3.1], [4.8, 2.7],
     [4.5, 2.5], [7.4, 2.9],
     [5.1, 2.7], [6.9, 2.7], [7.6, 2.6], [9.0, 5.0], [7.1, 2.2], [5.0, 2.7], [5.6, 2.4], [3.6, 4.8],
     [6.0, 2.4], [6.9, 2.9],
     [8.3, 4.9], [3.9, 4.0], [4.9, 3.1], [8.7, 3.9], [6.3, 2.4], [6.8, 2.5], [5.8, 2.1], [4.5, 4.1],
     [4.7, 3.2], [6.3, 2.6],
     [8.8, 4.8], [8.6, 4.1], [4.5, 3.8], [3.6, 4.3], [8.8, 5.0], [4.2, 3.9], [8.6, 4.4], [8.8, 4.0],
     [5.0, 3.4], [6.4, 2.5],
     [4.6, 2.6], [6.0, 2.6], [8.1, 3.5], [8.7, 4.5], [4.8, 2.8], [5.9, 2.7], [6.8, 2.6], [8.9, 4.6],
     [6.4, 2.6], [6.9, 2.5],
     [8.8, 3.3], [3.7, 4.0], [8.3, 4.0], [3.6, 4.3], [7.2, 2.2], [8.8, 4.4], [8.7, 4.7], [3.8, 4.4],
     [8.1, 3.4], [3.5, 4.7],
     [8.7, 4.1], [4.3, 3.8], [3.6, 4.0], [5.0, 2.7], [7.7, 3.2], [8.4, 3.2], [4.3, 3.7], [8.6, 4.3],
     [7.5, 3.2], [8.3, 3.8],
     [4.9, 2.9], [5.4, 2.4], [3.9, 4.9], [8.9, 3.6], [8.3, 3.4], [8.2, 3.3], [7.8, 2.8], [8.2, 3.2],
     [8.9, 4.8], [8.6, 3.8],
     [3.9, 5.3], [4.4, 4.6], [7.8, 3.0], [6.9, 2.7], [7.7, 3.0], [3.7, 3.7], [6.6, 3.0], [5.3, 2.6],
     [4.4, 4.1], [8.1, 3.6],
     [8.5, 3.4], [8.0, 3.7], [5.2, 2.7], [7.3, 2.8], [4.1, 4.0], [8.5, 3.6], [7.5, 2.4], [3.9, 3.8],
     [5.9, 2.5], [6.6, 2.9],
     [4.4, 3.4], [4.8, 3.3], [4.4, 3.1], [8.7, 4.8], [6.2, 2.7], [5.0, 3.2], [5.6, 2.7], [8.5, 4.2],
     [4.2, 3.5], [4.0, 3.1],
     [3.8, 4.1], [5.3, 2.2], [4.9, 3.3], [5.7, 3.1], [4.4, 3.5], [5.3, 2.8], [4.2, 3.3], [8.4, 3.6],
     [8.1, 3.5], [3.8, 4.4],
     [3.6, 4.3], [4.3, 4.6], [7.9, 3.1], [8.9, 4.9], [7.8, 3.2], [4.1, 3.7], [4.8, 3.1], [3.7, 4.3],
     [8.5, 3.8], [5.2, 2.7],
     [7.3, 2.8], [6.5, 2.6], [8.4, 4.3], [8.2, 4.0], [7.2, 2.9], [3.7, 4.2], [7.6, 2.6], [4.3, 4.7],
     [4.5, 3.5], [4.0, 4.2],
     [6.4, 2.7], [6.3, 2.6], [8.9, 3.9], [5.8, 2.3], [6.1, 2.6], [4.1, 3.7], [8.2, 3.1], [9.1, 4.5],
     [3.7, 4.1], [6.3, 2.7]])

# 将 class1_points 分割为训练集和测试集  
np.random.shuffle(class1_points)  # 随机打乱数据  
split_index = int(0.1 * len(class1_points))  # 取前10%的数据作为测试集  

# 将 class1 和 class2 中的数据分为训练和测试集  
class1_train_points = class1_points[split_index:]  # 90%的 class1 数据作为训练集  
class2_train_points = class2_points[split_index:]  # 90%的 class2 数据作为训练集  
class1_test_points = class1_points[:split_index]    # 10%的 class1 数据作为测试集  
class2_test_points = class2_points[:split_index]    # 10%的 class2 数据作为测试集  

# 合并训练集  
train_points = np.concatenate((class1_train_points, class2_train_points))  # 合并两个类别的训练点  
# 创建训练标签，类别1用0表示，类别2用1表示  
train_labels1 = np.zeros(len(class1_train_points))  # 类别1的标签  
train_labels2 = np.ones(len(class2_train_points))    # 类别2的标签  
train_labels = np.concatenate((train_labels1, train_labels2))  # 合并所有训练标签  

# 合并测试集  
test_points = np.concatenate((class1_test_points, class2_test_points))  # 合并两个类别的测试点  
# 创建测试标签  
test_labels1 = np.zeros(len(class1_test_points))  # 类别1的标签  
test_labels2 = np.ones(len(class2_test_points))    # 类别2的标签  
test_labels = np.concatenate((test_labels1, test_labels2))  # 合并所有测试标签  

# 2. 定义前向模型  
class YourModelClass(nn.Module):  
    def __init__(self):  
        super(YourModelClass, self).__init__()  
        # 定义六层的全连接神经网络结构  
        self.layer1 = nn.Linear(2, 8)   # 输入层到第一隐藏层  
        self.layer2 = nn.Linear(8, 16)  # 第一隐藏层到第二隐藏层  
        self.layer3 = nn.Linear(16, 32) # 第二隐藏层到第三隐藏层  
        self.layer4 = nn.Linear(32, 16) # 第三隐藏层到第四隐藏层  
        self.layer5 = nn.Linear(16, 8)  # 第四隐藏层到第五隐藏层  
        self.layer6 = nn.Linear(8, 2)   # 第五隐藏层到输出层  

    def forward(self, x):  
        # 前向传播函数  
        x = torch.tanh(self.layer1(x))  # 使用tanh激活函数  
        x = torch.tanh(self.layer2(x))  
        x = torch.tanh(self.layer3(x))  
        x = torch.tanh(self.layer4(x))  
        x = torch.tanh(self.layer5(x))  
        x = torch.softmax(self.layer6(x), dim=1)  # 使用softmax激活函数进行分类  
        return x  

# 实例化模型  
model = YourModelClass()  

# 3. 定义损失函数和优化器  
criterion = nn.CrossEntropyLoss()  # 交叉熵损失用于多分类问题  
optimizer = optim.Adam(model.parameters(), lr=0.01, weight_decay=0.005)  # Adam优化器，学习率和权重衰减  

# 创建等高线绘图的网格点  
x_min, x_max = 0, 10  
y_min, y_max = 0, 10  
step_size = 0.2  
xx, yy = np.meshgrid(np.arange(x_min, x_max, step_size), np.arange(y_min, y_max, step_size))  # 生成网格点  
grid_points = np.c_[xx.ravel(), yy.ravel()]  # 将网格点展平为二维数组  

# 创建图形和子图  
fig = plt.figure(figsize=(10, 5))  

ax1 = fig.add_subplot(121)  # 左侧图  
ax2 = fig.add_subplot(122)  # 右侧图  

step_list = []       # 存储训练步数  
loss_list = []       # 存储训练损失  
test_step_list = []  # 存储测试步数  
test_loss_list = []  # 存储测试损失  

# 4. 开始迭代  
num_iterations = 2000  
for n in range(num_iterations + 1):  
    # 将numpy数据转换为torch tensor  
    inputs = torch.tensor(train_points, dtype=torch.float32)  # 将训练点转换为张量  
    train_labels = torch.tensor(train_labels, dtype=torch.long)  # 将训练标签转换为张量  

    # 前向传播  
    outputs = model(inputs)  # 得到模型输出  
    loss = criterion(outputs, train_labels)  # 计算损失  

    # 反向传播和优化  
    optimizer.zero_grad()  # 清除梯度  
    loss.backward()        # 反向传播计算梯度  
    optimizer.step()       # 更新参数  

    # 更新损失图数据  
    step_list.append(n)    # 记录当前步数  
    loss_list.append(loss.detach())  # 记录当前损失值  

    # 5. 显示频率设置  
    frequency_display = 50  # 每50步输出一次信息  
    # 6. 显示与输出  
    if n % frequency_display == 0 or n == 1:  
        # 使用训练好的模型预测网格点的标签  
        grid_points_tensor = torch.tensor(grid_points, dtype=torch.float32)  # 将网格点转换为张量  
        Z = model(grid_points_tensor).detach().numpy()  # 得到予测输出  
        Z = Z[:, 1]  # 取类别2的概率值（1的列）  
        Z = Z.reshape(xx.shape)  # 调整Z的形状以适应网格  

        # 绘制2D图形  
        ax1.clear()  # 清除当前图  
        ax1.scatter(class1_train_points[:, 0], class1_train_points[:, 1], c='blue', label='label1')  # 类别1的点  
        ax1.scatter(class2_train_points[:, 0], class2_train_points[:, 1], c='red', label='label2')    # 类别2的点  
        ax1.contour(xx, yy, Z, levels=[0.5], colors='black')  # 绘制等高线  

        # 计算测试集损失  
        test_inputs = torch.tensor(test_points, dtype=torch.float32)  # 将测试点转换为张量  
        y_pred_test = model(test_inputs)  # 得到模型输出  
        test_labels = torch.tensor(test_labels, dtype=torch.long)  # 将测试标签转换为张量  
        loss_test = criterion(y_pred_test, test_labels)  # 计算测试集损失  
        test_step_list.append(n)  # 记录测试步数  
        test_loss_list.append(loss_test.detach())  # 记录测试损失  

        ax2.clear()  # 清除当前损失图  
        ax2.plot(step_list, loss_list, 'r-', label='Train Loss')  # 绘制训练损失  
        ax2.plot(test_step_list, test_loss_list, 'b-', label='Test Loss')  # 绘制测试损失  
        ax2.set_xlabel("Step")  # x轴标签  
        ax2.set_ylabel("Loss")  # y轴标签  
        ax2.legend()  # 显示图例  

plt.show()  # 展示图形