오늘은

개요

\frac{\mathrm{d}y}{\mathrm{d}x} = y(1-y)

y=\sigma(x)=\frac{1}{1+\exp(-x)}

구현 예

import numpy as np
from matplotlib import pyplot as plt


def get_integral_curve(f, init_xy, x_end, delta):
    """
    solve ode 'dy/dx=f(x,y)' with Euler method
    """
    (x, y) = init_xy
    xs, ys = [x], [y]
    for i in np.arange(init_xy[0], x_end, delta):
        y += delta*f(x, y)
        x += delta
        xs.append(x)
        ys.append(y)
    return xs, ys


def main():
    """
    solve logistic equation dy/dx=y(1-y)
    """
    # set parameters
    init_xy = (0, 0.5)
    x_end = 5
    delta = 0.1
    # calc integral curve of logistic equation
    f = lambda x, y: y*(1-y)
    xs, ys = get_integral_curve(f, init_xy, x_end, delta)
    # calc explicit solution
    sigmoid_xs = np.arange(-5, 5, delta)
    sigmoid_ys = 1/(1+np.exp(-sigmoid_xs))
    # plot
    fig, ax = plt.subplots()
    ax.plot(xs, ys, "x", color='blue', label="integral curve")
    ax.plot(sigmoid_xs, sigmoid_ys, "-", color='red', label='explicit curve')
    ax.legend()
    ax.set_xlim([-5, 5])
    ax.set_ylim([-0.5, 1.5])
    plt.show()

if __name__ == '__main__':
    main()

동작 예