import torch
import torch.nn as nn
import torch.optim as optim
import torch.nn.functional as F

class CubicSplineInterpolation(nn.Module):
    def __init__(self, data_points, derivatives):
        super(CubicSplineInterpolation, self).__init__()
        self.data_points = data_points
        self.derivatives = derivatives

    def forward(self, x):
        n = self.data_points.size(0)
        h = self.data_points[1:] - self.data_points[:-1]
        alpha = (3 * (self.derivatives[1:] - self.derivatives[:-1]) / h 
                 - 3 * (self.data_points[1:] - self.data_points[:-1]) / h.pow(2))
        beta = (2 * (self.data_points[:-1] - self.data_points[1:]) / h.pow(3) 
                - (self.derivatives[:-1] + 2 * self.derivatives[1:]) / h)
        s = torch.zeros_like(x)
        for i in range(n - 1):
            mask = (self.data_points[i] <= x) & (x < self.data_points[i+1])
            s[mask] = (self.derivatives[i] * (self.data_points[i+1] - x[mask]).pow(3) / (6 * h[i])
                       + self.derivatives[i+1] * (x[mask] - self.data_points[i]).pow(3) / (6 * h[i])
                       + alpha[i] * (self.data_points[i+1] - x[mask]) / (6 * h[i])
                       + beta[i] * (x[mask] - self.data_points[i]) / (6 * h[i]))
        return s

# 定义离散数据点和导数
data_points = torch.tensor([0.0, 1.0, 2.0, 3.0])
derivatives = torch.tensor([1.0, 2.0, 0.5, -1.0])

# 创建三次样条插值对象
interpolation = CubicSplineInterpolation(data_points, derivatives)

# 调用插值函数计算任意点上的曲线值
x = torch.tensor([0.5, 1.5, 2.5])
s = interpolation(x)
print(s)

tensor([ 1.0417,  0.6875, -0.3333])

class Integration(nn.Module):
    def __init__(self, function):
        super(Integration, self).__init__()
        self.function = function

    def forward(self, a, b, n):
        x = torch.linspace(a, b, n+1)
        y = self.function(x)
        h = (b - a) / n
        area = (h / 2) * (y[0] + 2 * y[1:n] + y[n])
        return area

# 定义待积分的函数
def f(x):
    return x**2

# 创建积分对象
integration = Integration(f)

# 调用积分函数计算函数在给定区间上的面积
area = integration(0, 1, 100)
print(area)

tensor(0.3333)