寻找非线性方程python scipy

Question

I am new to analytics , I am looking for a solution to find a model to solve non linear equation of the form Y=a(X1^b) + c(X2^d) + e ( where X1 , X2 are independent variables) Below is a full set , unfortunately we dont have much observations , all we need is any simple fit.But this data does not have any outliers , every observations has to be considered.

import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import numpy as np
import sympy as sym

x1=np.array([217,160,97,75])
x2=np.array([5.319751464,6.88536761,5.319751464,5.319751464])
x3=np.array([143.420635344,36.7384534912,23.420635344,1.420635344])
y=np.array([14,7,7,1])

def func(X, a, b, c ,d , e ):
    x1,x2 = X
    return a*x1**b + c*x2**d + e

popt, pcov = curve_fit(func, (x1,x2), y)

plt.plot(y, func((x1,x2), *popt), label="Fitted Curve")
plt.legend(loc='upper left')
plt.show()

but running curve_fit gives me error saying that the

TypeError: Improper input: N=5 must not exceed M=4

Then i had to add few more dummy inputs as observations of almost similar values adding decimal point difference which resulted in error

x1=np.array([217,160,97,75,76,219])
x2=np.array([5.319751464,6.88536761,5.319751464,5.319751464,5.319751464,5.319751464])
x3=np.array([143.420635344,36.7384534912,23.420635344,1.420635344,1.420635344,143.420635344])
y=np.array([14,7,7,1,1,14])

RuntimeError: Optimal parameters not found: Number of calls to function has reached maxfev = 1200.

Then I had to remove variable d and keep function as

def func(X, a, b, c ,e ):
    x1,x2 = X
    return a*x1**b + c*x2 + e

Finally it did run but again with below warning , but results are not good

RuntimeWarning: overflow encountered in power

Note that

x3 = max(x2 - {(x1^2)*2.6},0)

and solving

y=a*(x3^b) gives a=0.89 and b=0.58 with r2=0.98 and error=0.19 which is the best one i could get so far

But i would like to have the result in a generalised form without me trying to equate a relation. Because based on data set ,the function x3=f(x1,x2) can change and it is not a fixed equation for all cases.

Answer 1

A very good approach to get acceptable fit results is the use of initial guesses of the free parameters. curve_fit takes a p0 argument. If your data does not vary too much between two data sets, this should be a proper way. In my experience, taking some afford to guess good start values p0 and set limits by the parameter bounds is a good approach. Otherwise you could try to interpolate the data, if you don't have to know the mathematical relation.