为什么我的曲线拟合拟合对初始猜测参数如此敏感
why is my fitting with curve fit so sensitive to initial guess parameters
我知道，有很多例子，人们试图用曲线拟合来拟合 cos 但失败了。
可是看了很多，还是没头绪，我的脚本是怎么回事
数据比较清楚。我的代码有时确实有效，但不可靠。
这是我的最小示例：
import matplotlib.pyplot as plt
from numpy import cos, pi, inf
from numpy import diag as ndiag
from numpy import sqrt as nsqrt
from scipy.optimize import curve_fit
from scipy.signal import find_peaks
def test(power, transmission):
# function for p0
def find_initial_guess(power, transmission):
# find extrema for estimating initial guess
# skip minor minima, which are just noise bumps
minima, _ = find_peaks([-t for t in transmission], width=len(transmission) / 10)
maxima, _ = find_peaks(transmission, width=len(transmission) / 10)
# find neighbouring maxima and minima
# select arbitrary maxima from the middle of the list ( so it is not just an edge case
maximum = maxima[int(len(maxima) / 2)]
minimum = min([maximum - m for m in minima])
# distance of minimum and maximum can be used as initial guess for par2
return [sum(transmission) / (len(transmission)), # offset is average of transmission
(max(transmission) - min(transmission)) / 2, # amplitude is half the max-min-difference
pi, # phase offset
pi / (abs(power[maximum] - power[minimum])), # periodicity is 1 / distance(max-min)
] #
# fit function
def thermo_optic_coupling(p_heater, offset, amplitude, par1, par2):
return offset + amplitude * cos((par1 + par2 * p_heater))
initial_guess_popt = find_initial_guess(power, transmission)
popt, pcov = curve_fit(thermo_optic_coupling,
power,
transmission,
p0=initial_guess_popt,
bounds=[(-inf, 0, -6 * pi, -inf),
(inf, inf, 6 * pi, inf)],
maxfev=5000
)
popt = list(popt.tolist())
print('fit :', popt)
print('cov :', pcov)
print('summed root mean square error', sum(nsqrt(ndiag(pcov))))
fit = [thermo_optic_coupling(p_i, *popt) for p_i in power]
initial_guess = [thermo_optic_coupling(p_i, *initial_guess_popt) for p_i in power]
plt.plot(power, fit, color='orange', label='fit by scipy')
plt.plot(power, initial_guess, color='red', label='initial guess')
plt.plot(power, transmission, 'b+', label='experiment')
plt.xlabel('heater power ∝ V²')
plt.legend()
plt.ylabel('transmission in mW')
plt.show()
power1, transmission1 = [
[0.0, 1.5625, 3.1250000000000004, 4.687500000000001, 6.25, 7.812500000000001, 9.375, 10.9375,
12.500000000000002, 14.0625, 15.625, 17.187500000000004, 18.750000000000004, 20.3125,
21.875000000000004, 23.4375, 25.0, 26.562500000000004, 28.125000000000004,
29.687500000000004, 31.250000000000004, 32.81249999999999, 34.37500000000001, 35.9375, 37.5,
39.0625, 40.625, 42.1875, 43.75, 45.3125, 46.87500000000001, 48.43749999999999,
50.00000000000001, 51.5625, 53.125, 54.6875, 56.25, 57.8125, 59.375, 60.93749999999999,
62.5, 64.0625, 65.62499999999999, 67.1875, 68.75000000000001, 70.31249999999999, 71.875,
73.4375, 75.00000000000001, 76.5625, 78.125, 79.68749999999999, 81.25, 82.81250000000001,
84.37500000000001, 85.9375, 87.50000000000001, 89.06249999999999, 90.62500000000001,
92.1875, 93.75, 95.31249999999999, 96.875, 98.4375],
[1.6562295294369114e-06, 1.814877400886339e-06, 1.9415786380553917e-06,
2.032434337241163e-06, 2.07977581456966e-06, 2.0990197432469184e-06, 2.0990197432469184e-06,
2.0512402206219947e-06, 2.0045482927134235e-06, 1.9328632951064806e-06,
1.839538286344573e-06, 1.742643390948694e-06, 1.640984668617779e-06, 1.5322308981965023e-06,
1.3777010363167424e-06, 1.2493325696095153e-06, 1.1189650542693485e-06,
9.868258640760248e-07, 8.674581337360867e-07, 7.562624260369136e-07, 6.558233781566861e-07,
5.746673504788186e-07, 4.927385300164335e-07, 4.2957559516582234e-07, 3.70524153320037e-07,
3.5060372305501673e-07, 3.70524153320037e-07, 3.950456580084038e-07, 4.4306013694031485e-07,
5.232635985089236e-07, 6.256551763793945e-07, 7.528015828064533e-07, 8.754201193092522e-07,
1.033904733256672e-06, 1.1609580401117936e-06, 1.294566057151215e-06,
1.4211237849175981e-06, 1.539030734443489e-06, 1.648589477558925e-06,
1.7586133663896758e-06, 1.8561647616293735e-06, 1.9415786380553917e-06,
2.0045482927134235e-06, 2.0702201126993254e-06, 2.0990197432469184e-06,
2.07977581456966e-06, 2.0138008672355043e-06, 1.9415786380553917e-06, 1.814877400886339e-06,
1.6871454442159815e-06, 1.539030734443489e-06, 1.3655054776415766e-06,
1.1989926575358521e-06, 1.0195499872527435e-06, 8.363277813856736e-07,
6.614627809764817e-07, 4.950220262917964e-07, 3.554811066871505e-07, 2.3324724541289197e-07,
1.422237348203287e-07, 7.730111429536002e-08, 3.5477242527119533e-08,
2.6814427162334455e-08, 4.768628023291028e-08]]
# the data is small, so multiply transmission by 1e6
test(power1, [t_i * 1e6 for t_i in transmission1])
power2, transmission2 = [
[0.0, 1.5625, 3.1250000000000004, 4.687500000000001, 6.25, 7.812500000000001, 9.375, 10.9375, 12.500000000000002,
14.0625, 15.625, 17.187500000000004, 18.750000000000004, 20.3125, 21.875000000000004, 23.4375, 25.0,
26.562500000000004, 28.125000000000004, 29.687500000000004, 31.250000000000004, 32.81249999999999,
34.37500000000001, 35.9375, 37.5, 39.0625, 40.625, 42.1875, 43.75, 45.3125, 46.87500000000001, 48.43749999999999,
50.00000000000001, 51.5625, 53.125, 54.6875, 56.25, 57.8125, 59.375, 60.93749999999999, 62.5, 64.0625,
65.62499999999999, 67.1875, 68.75000000000001, 70.31249999999999, 71.875, 73.4375, 75.00000000000001, 76.5625,
78.125, 79.68749999999999, 81.25, 82.81250000000001, 84.37500000000001, 85.9375, 87.50000000000001,
89.06249999999999, 90.62500000000001, 92.1875, 93.75, 95.31249999999999, 96.875, 98.4375],
[1.396174174553104e-06, 1.2437906644035389e-06, 1.0933734492030595e-06, 9.231950062379674e-07,
7.493565772474859e-07, 5.942121443423389e-07, 4.470013463798467e-07, 3.209402716365006e-07, 2.0673619727046636e-07,
1.2646679820627658e-07, 6.225003507478826e-08, 2.7063535041190432e-08, 1.5904518423724243e-08,
3.301448473686211e-08, 6.953709180130387e-08, 1.3921171470244173e-07, 2.3002252671138602e-07,
3.5060372305501673e-07, 4.927385300164335e-07, 6.586370438602071e-07, 8.556507267350914e-07, 1.038734421672639e-06,
1.2045269561883742e-06, 1.3295627259399386e-06, 1.4593866518595935e-06, 1.5741007416815629e-06,
1.6793627797419938e-06, 1.7507193165159197e-06, 1.823060741632163e-06, 1.8983913848020828e-06,
1.9155497976900293e-06, 1.8983913848020828e-06, 1.847832823818452e-06, 1.7665430107889611e-06,
1.633414940040678e-06, 1.4920581674760974e-06, 1.3295627259399386e-06, 1.1717002525967601e-06,
1.024312620215478e-06, 8.635043377359954e-07, 7.092257962431407e-07, 5.480251928284404e-07, 3.897772826233099e-07,
2.6497879338257794e-07, 1.584588586826325e-07, 9.344973238413863e-08, 5.650105063022198e-08, 5.224160649199307e-08,
7.561515230224364e-08, 1.2531196020926962e-07, 2.001284497590038e-07, 3.0707095931959394e-07,
4.352970804255852e-07, 5.857557001760154e-07, 7.632319170881868e-07, 9.480514529881339e-07, 1.129318710440296e-06,
1.3061280439519783e-06, 1.4593866518595935e-06, 1.6034836040919702e-06, 1.7186384491686294e-06,
1.814877400886339e-06, 1.847832823818452e-06, 1.872941513580031e-06]]
# the data is small, so multiply transmission by 1e6
test(power2, [t_i * 1e6 for t_i in transmission2])
power3, transmission3 = [
[0.0, 1.5625, 3.1250000000000004, 4.687500000000001, 6.25, 7.812500000000001, 9.375, 10.9375, 12.500000000000002,
14.0625, 15.625, 17.187500000000004, 18.750000000000004, 20.3125, 21.875000000000004, 23.4375, 25.0,
26.562500000000004, 28.125000000000004, 29.687500000000004, 31.250000000000004, 32.81249999999999,
34.37500000000001, 35.9375, 37.5, 39.0625, 40.625, 42.1875, 43.75, 45.3125, 46.87500000000001, 48.43749999999999,
50.00000000000001, 51.5625, 53.125, 54.6875, 56.25, 57.8125, 59.375, 60.93749999999999, 62.5, 64.0625,
65.62499999999999, 67.1875, 68.75000000000001, 70.31249999999999, 71.875, 73.4375, 75.00000000000001, 76.5625,
78.125, 79.68749999999999, 81.25, 82.81250000000001, 84.37500000000001, 85.9375, 87.50000000000001,
89.06249999999999, 90.62500000000001, 92.1875, 93.75, 95.31249999999999, 96.875, 98.4375],
[1.663904987577944e-06, 1.8312809813253925e-06, 1.959127394435598e-06, 2.0512402206219947e-06,
2.089375623555793e-06, 2.0990197432469184e-06, 2.128220016564131e-06, 2.1087083781719565e-06,
2.0512402206219947e-06, 2.0045482927134235e-06, 1.9155497976900293e-06, 1.7986207544535706e-06,
1.6871454442159815e-06, 1.552721071155355e-06, 1.414844892100276e-06, 1.294566057151215e-06,
1.1503143128228655e-06, 1.038734421672639e-06, 9.069871600668335e-07, 7.989811320185081e-07, 7.027494648298712e-07,
6.203210568077912e-07, 5.330310688570652e-07, 4.7348784131211617e-07, 4.4502637870081293e-07,
4.2580311206559627e-07, 4.2957559516582234e-07, 4.6311990409702625e-07, 4.905614836185237e-07,
5.66489055218778e-07, 6.558233781566861e-07, 7.562624260369136e-07, 8.754201193092522e-07, 1.0148094985771049e-06,
1.1556239224450914e-06, 1.2717482214298512e-06, 1.3777010363167424e-06, 1.4986797225052762e-06,
1.6034836040919702e-06, 1.7266031287264504e-06, 1.814877400886339e-06, 1.9241870735205803e-06,
1.9953382299096967e-06, 2.0702201126993254e-06, 2.0990197432469184e-06, 2.089375623555793e-06,
2.041815627896478e-06, 1.959127394435598e-06, 1.8561647616293735e-06, 1.7266031287264504e-06,
1.5813955906972397e-06, 1.4023702092175955e-06, 1.2493325696095153e-06, 1.0731798840385246e-06,
8.832073557363544e-07, 7.124863091749079e-07, 5.355012926902759e-07, 3.791547695631538e-07, 2.5433475611234004e-07,
1.5230712743292288e-07, 8.20050359899154e-08, 3.9698159260983884e-08, 2.744154275033995e-08,
4.559654207832307e-08]]
# the data is small, so multiply transmission by 1e6
test(power3, [t_i * 1e6 for t_i in transmission3])
有谁知道我如何为拟合提供更多参数，使其对初始猜测参数不那么敏感？
为什么 scipy 不能可靠地调整相位偏移？
提前致谢，
菲利克斯
首先，规范化数据是个好主意。那么，你的界限就很大了。以您的示例为例，我固定了靠近您的 'guess' 的边界并增加了 maxfex:
bounds=[(-10*initial_guess_popt[0], -10*initial_guess_popt[1], 0.75*initial_guess_popt[2], -10*initial_guess_popt[3]),
(10*initial_guess_popt[0], 10*initial_guess_popt[1], 1.25*initial_guess_popt[2], 10*initial_guess_popt[3])],
maxfev=50000
)
我知道，有很多例子，人们试图用曲线拟合来拟合 cos 但失败了。
可是看了很多，还是没头绪，我的脚本是怎么回事
数据比较清楚。我的代码有时确实有效，但不可靠。
这是我的最小示例：
import matplotlib.pyplot as plt
from numpy import cos, pi, inf
from numpy import diag as ndiag
from numpy import sqrt as nsqrt
from scipy.optimize import curve_fit
from scipy.signal import find_peaks
def test(power, transmission):
# function for p0
def find_initial_guess(power, transmission):
# find extrema for estimating initial guess
# skip minor minima, which are just noise bumps
minima, _ = find_peaks([-t for t in transmission], width=len(transmission) / 10)
maxima, _ = find_peaks(transmission, width=len(transmission) / 10)
# find neighbouring maxima and minima
# select arbitrary maxima from the middle of the list ( so it is not just an edge case
maximum = maxima[int(len(maxima) / 2)]
minimum = min([maximum - m for m in minima])
# distance of minimum and maximum can be used as initial guess for par2
return [sum(transmission) / (len(transmission)), # offset is average of transmission
(max(transmission) - min(transmission)) / 2, # amplitude is half the max-min-difference
pi, # phase offset
pi / (abs(power[maximum] - power[minimum])), # periodicity is 1 / distance(max-min)
] #
# fit function
def thermo_optic_coupling(p_heater, offset, amplitude, par1, par2):
return offset + amplitude * cos((par1 + par2 * p_heater))
initial_guess_popt = find_initial_guess(power, transmission)
popt, pcov = curve_fit(thermo_optic_coupling,
power,
transmission,
p0=initial_guess_popt,
bounds=[(-inf, 0, -6 * pi, -inf),
(inf, inf, 6 * pi, inf)],
maxfev=5000
)
popt = list(popt.tolist())
print('fit :', popt)
print('cov :', pcov)
print('summed root mean square error', sum(nsqrt(ndiag(pcov))))
fit = [thermo_optic_coupling(p_i, *popt) for p_i in power]
initial_guess = [thermo_optic_coupling(p_i, *initial_guess_popt) for p_i in power]
plt.plot(power, fit, color='orange', label='fit by scipy')
plt.plot(power, initial_guess, color='red', label='initial guess')
plt.plot(power, transmission, 'b+', label='experiment')
plt.xlabel('heater power ∝ V²')
plt.legend()
plt.ylabel('transmission in mW')
plt.show()
power1, transmission1 = [
[0.0, 1.5625, 3.1250000000000004, 4.687500000000001, 6.25, 7.812500000000001, 9.375, 10.9375,
12.500000000000002, 14.0625, 15.625, 17.187500000000004, 18.750000000000004, 20.3125,
21.875000000000004, 23.4375, 25.0, 26.562500000000004, 28.125000000000004,
29.687500000000004, 31.250000000000004, 32.81249999999999, 34.37500000000001, 35.9375, 37.5,
39.0625, 40.625, 42.1875, 43.75, 45.3125, 46.87500000000001, 48.43749999999999,
50.00000000000001, 51.5625, 53.125, 54.6875, 56.25, 57.8125, 59.375, 60.93749999999999,
62.5, 64.0625, 65.62499999999999, 67.1875, 68.75000000000001, 70.31249999999999, 71.875,
73.4375, 75.00000000000001, 76.5625, 78.125, 79.68749999999999, 81.25, 82.81250000000001,
84.37500000000001, 85.9375, 87.50000000000001, 89.06249999999999, 90.62500000000001,
92.1875, 93.75, 95.31249999999999, 96.875, 98.4375],
[1.6562295294369114e-06, 1.814877400886339e-06, 1.9415786380553917e-06,
2.032434337241163e-06, 2.07977581456966e-06, 2.0990197432469184e-06, 2.0990197432469184e-06,
2.0512402206219947e-06, 2.0045482927134235e-06, 1.9328632951064806e-06,
1.839538286344573e-06, 1.742643390948694e-06, 1.640984668617779e-06, 1.5322308981965023e-06,
1.3777010363167424e-06, 1.2493325696095153e-06, 1.1189650542693485e-06,
9.868258640760248e-07, 8.674581337360867e-07, 7.562624260369136e-07, 6.558233781566861e-07,
5.746673504788186e-07, 4.927385300164335e-07, 4.2957559516582234e-07, 3.70524153320037e-07,
3.5060372305501673e-07, 3.70524153320037e-07, 3.950456580084038e-07, 4.4306013694031485e-07,
5.232635985089236e-07, 6.256551763793945e-07, 7.528015828064533e-07, 8.754201193092522e-07,
1.033904733256672e-06, 1.1609580401117936e-06, 1.294566057151215e-06,
1.4211237849175981e-06, 1.539030734443489e-06, 1.648589477558925e-06,
1.7586133663896758e-06, 1.8561647616293735e-06, 1.9415786380553917e-06,
2.0045482927134235e-06, 2.0702201126993254e-06, 2.0990197432469184e-06,
2.07977581456966e-06, 2.0138008672355043e-06, 1.9415786380553917e-06, 1.814877400886339e-06,
1.6871454442159815e-06, 1.539030734443489e-06, 1.3655054776415766e-06,
1.1989926575358521e-06, 1.0195499872527435e-06, 8.363277813856736e-07,
6.614627809764817e-07, 4.950220262917964e-07, 3.554811066871505e-07, 2.3324724541289197e-07,
1.422237348203287e-07, 7.730111429536002e-08, 3.5477242527119533e-08,
2.6814427162334455e-08, 4.768628023291028e-08]]
# the data is small, so multiply transmission by 1e6
test(power1, [t_i * 1e6 for t_i in transmission1])
power2, transmission2 = [
[0.0, 1.5625, 3.1250000000000004, 4.687500000000001, 6.25, 7.812500000000001, 9.375, 10.9375, 12.500000000000002,
14.0625, 15.625, 17.187500000000004, 18.750000000000004, 20.3125, 21.875000000000004, 23.4375, 25.0,
26.562500000000004, 28.125000000000004, 29.687500000000004, 31.250000000000004, 32.81249999999999,
34.37500000000001, 35.9375, 37.5, 39.0625, 40.625, 42.1875, 43.75, 45.3125, 46.87500000000001, 48.43749999999999,
50.00000000000001, 51.5625, 53.125, 54.6875, 56.25, 57.8125, 59.375, 60.93749999999999, 62.5, 64.0625,
65.62499999999999, 67.1875, 68.75000000000001, 70.31249999999999, 71.875, 73.4375, 75.00000000000001, 76.5625,
78.125, 79.68749999999999, 81.25, 82.81250000000001, 84.37500000000001, 85.9375, 87.50000000000001,
89.06249999999999, 90.62500000000001, 92.1875, 93.75, 95.31249999999999, 96.875, 98.4375],
[1.396174174553104e-06, 1.2437906644035389e-06, 1.0933734492030595e-06, 9.231950062379674e-07,
7.493565772474859e-07, 5.942121443423389e-07, 4.470013463798467e-07, 3.209402716365006e-07, 2.0673619727046636e-07,
1.2646679820627658e-07, 6.225003507478826e-08, 2.7063535041190432e-08, 1.5904518423724243e-08,
3.301448473686211e-08, 6.953709180130387e-08, 1.3921171470244173e-07, 2.3002252671138602e-07,
3.5060372305501673e-07, 4.927385300164335e-07, 6.586370438602071e-07, 8.556507267350914e-07, 1.038734421672639e-06,
1.2045269561883742e-06, 1.3295627259399386e-06, 1.4593866518595935e-06, 1.5741007416815629e-06,
1.6793627797419938e-06, 1.7507193165159197e-06, 1.823060741632163e-06, 1.8983913848020828e-06,
1.9155497976900293e-06, 1.8983913848020828e-06, 1.847832823818452e-06, 1.7665430107889611e-06,
1.633414940040678e-06, 1.4920581674760974e-06, 1.3295627259399386e-06, 1.1717002525967601e-06,
1.024312620215478e-06, 8.635043377359954e-07, 7.092257962431407e-07, 5.480251928284404e-07, 3.897772826233099e-07,
2.6497879338257794e-07, 1.584588586826325e-07, 9.344973238413863e-08, 5.650105063022198e-08, 5.224160649199307e-08,
7.561515230224364e-08, 1.2531196020926962e-07, 2.001284497590038e-07, 3.0707095931959394e-07,
4.352970804255852e-07, 5.857557001760154e-07, 7.632319170881868e-07, 9.480514529881339e-07, 1.129318710440296e-06,
1.3061280439519783e-06, 1.4593866518595935e-06, 1.6034836040919702e-06, 1.7186384491686294e-06,
1.814877400886339e-06, 1.847832823818452e-06, 1.872941513580031e-06]]
# the data is small, so multiply transmission by 1e6
test(power2, [t_i * 1e6 for t_i in transmission2])
power3, transmission3 = [
[0.0, 1.5625, 3.1250000000000004, 4.687500000000001, 6.25, 7.812500000000001, 9.375, 10.9375, 12.500000000000002,
14.0625, 15.625, 17.187500000000004, 18.750000000000004, 20.3125, 21.875000000000004, 23.4375, 25.0,
26.562500000000004, 28.125000000000004, 29.687500000000004, 31.250000000000004, 32.81249999999999,
34.37500000000001, 35.9375, 37.5, 39.0625, 40.625, 42.1875, 43.75, 45.3125, 46.87500000000001, 48.43749999999999,
50.00000000000001, 51.5625, 53.125, 54.6875, 56.25, 57.8125, 59.375, 60.93749999999999, 62.5, 64.0625,
65.62499999999999, 67.1875, 68.75000000000001, 70.31249999999999, 71.875, 73.4375, 75.00000000000001, 76.5625,
78.125, 79.68749999999999, 81.25, 82.81250000000001, 84.37500000000001, 85.9375, 87.50000000000001,
89.06249999999999, 90.62500000000001, 92.1875, 93.75, 95.31249999999999, 96.875, 98.4375],
[1.663904987577944e-06, 1.8312809813253925e-06, 1.959127394435598e-06, 2.0512402206219947e-06,
2.089375623555793e-06, 2.0990197432469184e-06, 2.128220016564131e-06, 2.1087083781719565e-06,
2.0512402206219947e-06, 2.0045482927134235e-06, 1.9155497976900293e-06, 1.7986207544535706e-06,
1.6871454442159815e-06, 1.552721071155355e-06, 1.414844892100276e-06, 1.294566057151215e-06,
1.1503143128228655e-06, 1.038734421672639e-06, 9.069871600668335e-07, 7.989811320185081e-07, 7.027494648298712e-07,
6.203210568077912e-07, 5.330310688570652e-07, 4.7348784131211617e-07, 4.4502637870081293e-07,
4.2580311206559627e-07, 4.2957559516582234e-07, 4.6311990409702625e-07, 4.905614836185237e-07,
5.66489055218778e-07, 6.558233781566861e-07, 7.562624260369136e-07, 8.754201193092522e-07, 1.0148094985771049e-06,
1.1556239224450914e-06, 1.2717482214298512e-06, 1.3777010363167424e-06, 1.4986797225052762e-06,
1.6034836040919702e-06, 1.7266031287264504e-06, 1.814877400886339e-06, 1.9241870735205803e-06,
1.9953382299096967e-06, 2.0702201126993254e-06, 2.0990197432469184e-06, 2.089375623555793e-06,
2.041815627896478e-06, 1.959127394435598e-06, 1.8561647616293735e-06, 1.7266031287264504e-06,
1.5813955906972397e-06, 1.4023702092175955e-06, 1.2493325696095153e-06, 1.0731798840385246e-06,
8.832073557363544e-07, 7.124863091749079e-07, 5.355012926902759e-07, 3.791547695631538e-07, 2.5433475611234004e-07,
1.5230712743292288e-07, 8.20050359899154e-08, 3.9698159260983884e-08, 2.744154275033995e-08,
4.559654207832307e-08]]
# the data is small, so multiply transmission by 1e6
test(power3, [t_i * 1e6 for t_i in transmission3])
有谁知道我如何为拟合提供更多参数，使其对初始猜测参数不那么敏感？
为什么 scipy 不能可靠地调整相位偏移？
提前致谢， 菲利克斯
首先，规范化数据是个好主意。那么，你的界限就很大了。以您的示例为例，我固定了靠近您的 'guess' 的边界并增加了 maxfex:
bounds=[(-10*initial_guess_popt[0], -10*initial_guess_popt[1], 0.75*initial_guess_popt[2], -10*initial_guess_popt[3]),
(10*initial_guess_popt[0], 10*initial_guess_popt[1], 1.25*initial_guess_popt[2], 10*initial_guess_popt[3])],
maxfev=50000
)