I am using the quad function from scipy.integrate v0.19.1 to integrate functions with a square-root like singularity at each end of the integration interval such as for example

In [1]: quad(lambda x: 1/sqrt(1-x**2), -1, 1)

(I use the sqrt function from numpy v1.12.0) which immediately yields the correct result pi:

Out[1]: (3.141592653589591, 6.200897573194197e-10)

According to the documentation of the quad function the keyword points should be used to indicate the locations of singularities or discontinuities of the integrand, but …

我正在尝试使用Keras来建立一个神经网络，该网络应该近似一个未知函数F(x)。F(x)然后应使用所得的神经网络近似值来求和的最小值G(x) + F(x)，其中G(x)有一些任意函数。我当前面临的问题是，神经网络逼近F(x)不够平滑，因此局部优化器陷入了微小的局部极小值中。对于能改善结果或解决此问题的任何想法，我将深表感谢。

简单示例：最小化方差

让我说明上非常简单的例子问题：我会尽量教神经网络的功能F(x) = 4*var(x)与x = [x1,...,xN]和0 <= xi <= 1，这里var(x)是指矢量的变化x。随后，我将尝试查找F(x)约束条件下x具有给定均值的神经网络表示的最小值。此示例的完整代码可以在sec中找到。3以下。

1.神经网络的创建和训练

首先，我为以下近似创建一个神经网络F(x)：

N = 6  # Dimension of input vector x

# Set up the neural network
model = Sequential()
model.add(Dense(50, activation='sigmoid', input_dim=N))
model.add(Dense(1, activation='sigmoid'))
model.compile(optimizer='adam', loss='mse')

随后，我生成训练数据并训练模型：

# Generate training data
n_train = 100000  # …

为了熟悉全局优化方法，特别是shgo优化器，scipy.optimize v1.3.0我尝试在具有给定平均值的约束下最小化向量的方差var(x)：x = [x1,...,xN]0 <= xi <= 1x

import numpy as np
from scipy.optimize import shgo

# Constraint
avg = 0.5  # Given average value of x
cons = {'type': 'eq', 'fun': lambda x: np.mean(x)-avg}

# Minimize the variance of x under the given constraint
res = shgo(lambda x: np.var(x), bounds=6*[(0, 1)], constraints=cons)

该shgo方法在这个问题上失败了：

>>> res
     fun: 0.0
 message: 'Failed to find a feasible minimiser point. Lowest sampling point …