Why does curve_fit not fit 3d curve

I'm trying to fit a multiveriable curve to the function y**2*(y**2/4 - a*y/3 + b/2) - x**2*(x**2/4 - a*x/3 + b/2) but it seams that curve_fit doesn't fit to the generated data. The pcov values are all inf and the a and b outputs remain the same as the guesses.

Here is my code:

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


def func(X, a, b):
    x,y = X
    return y**2*(y**2/4 - a*y/3 + b/2) - x**2*(x**2/4 - a*x/3 + b/2)


# some artificially noisy data to fit

x = np.arange(0,5,0.1)
y = np.arange(0,5,0.1)
a = 6
b = 9
z = func((x,y), a, b)


# initial guesses for a,b:

popt, pcov = curve_fit(func, [x,y], z, p0=[10,8])

print(popt)
print(pcov)
print("a = ",popt[0] ," +/- ", np.sqrt(pcov[0,0]))
print("b = ",popt[1] ," +/- ", np.sqrt(pcov[1,1]))


#surface plot

ax = plt.axes(projection="3d")
x_data = np.arange(0,5,0.1)
y_data = np.arange(0,5,0.1)
X, Y = np.meshgrid(x_data,y_data)


Z = func((X,Y), popt[0], popt[1])
ax.scatter(X, Y, Z)
plt.show()

The code works for other functions but not this one.

Solution

When you increase the dimensionnality, you also have to sample correctly in all dimension. In your MCVE you are only sampling over a single line instead of sampling in the plane.

The following sampling spreads over the plane:

import numpy as np
import matplotlib.pyplot as plt
from scipy import optimize

def model(x, a, b):
    return x[1] ** 2 *(x[1] ** 2 / 4 - a * x[1] / 3 + b / 2) - x[0] ** 2 *(x[0] ** 2 / 4 - a * x[0] / 3 + b / 2)

x = np.arange(0, 5, 0.1)
y = np.arange(0, 5, 0.1)
X, Y = np.meshgrid(x, y)

a = 6
b = 9
x = (X.ravel(), Y.ravel())

z = model(x, a, b)

popt, pcov = optimize.curve_fit(model, x, z, p0=[10,8])
#(array([6., 9.]),
# array([[6.78893023e-33, 2.09471226e-32],
#        [2.09471226e-32, 6.64942515e-32]]))

And fits properly.

The following figure shows your sampling and the plane sampling with surface:

Also notice, than your original sampling lies exactly on single isopleth: