在Python中优化微分方程中的常数

问题描述

好吧，我将如何使用curve_fit来编写代码以优化微分方程中的常数a和b，例如dy/dt = a * y ^ 2 + b?我将使用odeint求解ODE，然后使用curve_fit优化a和b.如果您能提供有关这种情况的意见，我将不胜感激！

Okay so how would i approach to writing a code to optimize the constants a and b in a differential equation, like dy/dt = a*y^2 + b, using curve_fit? I would be using odeint to solve the ODE and then curve_fit to optimize a and b.If you could please provide input on this situation i would greatly appreciate it!

推荐答案

您绝对可以做到:

import numpy as np
from scipy.integrate import odeint
from scipy.optimize import curve_fit

def f(y, t, a, b):
    return a*y**2 + b

def y(t, a, b, y0):
    """
    Solution to the ODE y'(t) = f(t,y,a,b) with initial condition y(0) = y0
    """
    y = odeint(f, y0, t, args=(a, b))
    return y.ravel()

# Some random data to fit
data_t = np.sort(np.random.rand(200) * 10)
data_y = data_t**2 + np.random.rand(200)*10

popt, cov = curve_fit(y, data_t, data_y, [-1.2, 0.1, 0])
a_opt, b_opt, y0_opt = popt

print("a = %g" % a_opt)
print("b = %g" % b_opt)
print("y0 = %g" % y0_opt)

import matplotlib.pyplot as plt
t = np.linspace(0, 10, 2000)
plt.plot(data_t, data_y, '.',
         t, y(t, a_opt, b_opt, y0_opt), '-')
plt.gcf().set_size_inches(6, 4)
plt.savefig('out.png', dpi=96)
plt.show()

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