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Asked 1 month ago by SolarSatellite092

Why Does curve_fit Return Negative Parameters in the Debye–Callaway Thermal Conductivity Model?

The post content has been automatically edited by the Moderator Agent for consistency and clarity.

I'm trying to fit experimental thermal conductivity data into the Debye–Callaway equation, but one of the fitted parameters comes out negative despite trying different initial guesses.

I am using literature values (A = 6.73E-43, B = 5.38E-18) as a reference. In my first code, I obtain fitted parameters:
A_fit = 1.417070339751493e-44, B_fit = 1.075485244260637e-15, which are far from expectations. I also tried a second code version by adding an extra scattering parameter (C) in a new term, yet the second parameter still returns as negative. My concern is that the model may be insensitive to initial conditions or could be getting stuck in a local minimum.

Below is the first code snippet for the simple Debye–Callaway model:

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

# Constants
kb = 1.380649e-23  # Boltzmann constant (J/K)
hbar = 1.0545718e-34  # Reduced Planck's constant (J*s)
theta_D = 280  # Debye temperature (K)
v = 2300  # Average phonon group velocity (m/s)

# Define the inverse relaxation times
def tau_PD_inv(omega, A):
    return A * omega**4

def tau_U_inv(omega, B, T):
    return B * omega**2 * np.exp(-theta_D / (3 * T))

def tau_GB_inv(omega):
    D = 0.7e-6  # Grain boundary characteristic length (m)
    return v / D

def tau_total_inv(omega, A, B, T):
    return tau_PD_inv(omega, A) + tau_U_inv(omega, B, T) + tau_GB_inv(omega)

# Debye Callaway model for thermal conductivity
def kappa_lattice(T, A, B):
    def integrand(x):
        omega = x * kb * T / hbar
        tau_total = 1 / tau_total_inv(omega, A, B, T)
        return x**4 * np.exp(x) / (np.exp(x) - 1)**2 * tau_total

    integral, _ = quad(integrand, 0, theta_D / T)
    prefactor = kb**4 * T**3 / (2 * np.pi**2 * v * hbar**3)
    return prefactor * integral

# Vectorize the model for curve fitting
def kappa_lattice_vectorized(T, A, B ):
    return np.array([kappa_lattice(Ti, A, B ) for Ti in T])

# Example thermal conductivity and temperature data for Ge WQ
T_data = np.array([3.0748730372823156, 3.1962737114707203, 3.4441443065638015, 3.785528026732028,
    4.043999986888003, 4.524033859711473, 4.902329259251676, 5.270E+00,
    5.902534836759174, 6.721645553370004, 7.690683028568884, 9.031E+00,
    1.103E+01, 12.811754075061538, 16.07373842258417, 19.67675447459597,
    23.951202854256202, 29.710517518825778, 3.684E+01, 45.030707655028536,
    56.92469139605285, 6.929E+01, 85.62796640159988, 105.41803279520694,
    129.5096325800293, 159.77670578646953, 200.9314386671018, 240.34130109997898,
    291.63127697234717])  # Temperature in K
kappa_data = np.array([ 0.5021456742832973, 0.5907837911587943, 0.7331309897165834, 0.9570228747931655,
    1.1281866396926783, 1.4904675008427906, 1.8682223847710366, 2.1856825810901013,
    2.8515011626042894, 4.057791636291224, 5.328111104734368, 7.645627510566391,
    11.045409412114584, 14.277055520040301, 19.54065663430087, 24.738891215975602,
    28.776422188514683, 32.562485211039274, 33.37572711709385, 32.455084248250884,
    28.56802589740088, 24.937473590597776, 20.887192910896978, 17.37073097716312,
    14.783541083880793, 12.26010669954394, 9.991973525044475, 8.531955488387288,
    7.347906481962733])  # Thermal conductivity in W/mK

# Initial guess for parameters A, B
initial_guess = [4.75E-42, 1.054E-17]

# Curve fitting
popt, pcov = curve_fit(kappa_lattice_vectorized, T_data, kappa_data, p0=initial_guess, maxfev=8_000)
#popt, pcov = curve_fit(kappa_lattice_vectorized, T_data, kappa_data, p0=initial_guess, bounds=(0, np.inf))

# Extract fitted parameters
A_fit, B_fit = popt

# Print the fitted parameters
print(f"Fitted parameters: A = {A_fit}, B = {B_fit}")

# Generate curve fit data
T_fit = np.linspace(min(T_data), max(T_data), 100)
kappa_fit = kappa_lattice_vectorized(T_fit, *popt)

# Plot the original data, fitted data, and extrapolated data
plt.figure(figsize=(8, 6))
plt.plot(T_data, kappa_data, 'o', label='Experimental data')
plt.plot(T_fit, kappa_fit, '-', label='Fitted model')
#plt.plot(T_extrap, kappa_extrap, '--', label='Extrapolated (400-600K)')
plt.xlabel('Temperature (K)')
plt.ylabel('Thermal Conductivity (W/mK)')
plt.legend()
plt.title('Thermal Conductivity Fitting ')
plt.show()

And here is the second modified version using an extra parameter (C):

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

# Constants
kb = 1.380649e-23  # Boltzmann constant (J/K)
hbar = 1.0545718e-34  # Reduced Planck's constant (J*s)
theta_D = 483.0  # Debye temperature (K)
v = 4618.0  # Average phonon group velocity (m/s)

# Define the inverse relaxation times
def tau_PD_inv(omega, A):
    return A * omega**4

def tau_U_inv(omega, B, T):
    return B * omega**2 * T * np.exp(-theta_D / (3 * T))

def tau_GB_inv(omega):
    D = 1e-6  # Grain boundary characteristic length (m)
    return v / D

def tau_new_inv(omega, C):
    return C * omega**2

def tau_total_inv(omega, A, B, C, T):
    return tau_PD_inv(omega, A) + tau_U_inv(omega, B, T) + tau_GB_inv(omega) + tau_new_inv(omega, C)

# Debye Callaway model for thermal conductivity
def kappa_lattice(T, A, B, C):
    def integrand(x):
        omega = x * kb * T / hbar
        tau_total = 1 / tau_total_inv(omega, A, B, C, T)
        return x**4 * np.exp(x) / (np.exp(x) - 1)**2 * tau_total

    integral, _ = quad(integrand, 0, theta_D / T)
    prefactor = kb**4 * T**3 / (2 * np.pi**2 * v * hbar**3)
    return prefactor * integral

# Vectorize the model for curve fitting
def kappa_lattice_vectorized(T, A, B, C):
    return np.array([kappa_lattice(Ti, A, B, C) for Ti in T])

# Example thermal conductivity and temperature data for Ge WQ
T_data = np.array([31.21198032, 31.63863233, 33.88415583, 36.02698918, 38.39872854,
    39.56437842, 43.09065509, 44.50006634, 46.62080745, 43.20705304,
    47.77194375, 51.1374173, 53.33894737, 55.68342411, 57.7839973,
    60.44084481, 63.31137245, 66.34735373, 69.16983511, 74.09615366,
    77.63998236, 82.66616204, 87.95740211, 93.19748533, 99.59297955,
    105.6903923, 113.1334233, 120.6776065, 127.0279836, 134.1693093,
    141.3911575, 148.1783285, 155.5512459, 162.631427, 169.7085111,
    177.378102, 183.5898785, 190.9367188, 197.6917958, 205.7338436,
    212.8065032, 220.0698325, 226.7609477, 234.0571384, 241.1287763,
    247.8786187, 255.2722521, 262.3425521, 269.4163916, 276.4872816,
    283.5510925, 297.1613034, 304.7919793, 311.8668132, 318.9352276,
    326.0927208, 333.0893085, 340.164332, 347.2375209, 354.9542919,
    362.0246034, 369.0955138, 375.8247965, 383.5799572])  # Temperature in K
kappa_data = np.array([ 1.085444844, 1.13592882, 1.176429149, 1.217086899,
    1.258678341, 1.305290878, 1.357081371, 1.398857871, 1.26084173,
    1.436622858, 1.483042548, 1.528031021, 1.561551609, 1.598710119,
    1.632418606, 1.665656346, 1.703762245, 1.742520105, 1.774186181,
    1.820109169, 1.855169951, 1.893680154, 1.932568555, 1.962172074,
    1.999232142, 2.029595098, 2.0490363, 2.067859052, 2.092450212,
    2.119053495, 2.131413546, 2.145488003, 2.158842932, 2.169084281,
    2.169487956, 2.184424479, 2.196747698, 2.20620033, 2.217744549,
    2.223537925, 2.236483003, 2.258522251, 2.273748326, 2.278514432,
    2.28270469, 2.288247862, 2.291668986, 2.298648488, 2.302662675,
    2.299560106, 2.311410793, 2.339970141, 2.347949378, 2.349474819,
    2.354437976, 2.369869851, 2.378039597, 2.384365082, 2.39183344,
    2.395266095, 2.399300875, 2.40766296, 2.409888354, 2.442162261])  # Thermal conductivity in W/mK

# Initial guess for parameters A, B, and C
initial_guess = [4.75E-43, 3.6979E-19, 1e-16]

# Curve fitting
popt, pcov = curve_fit(kappa_lattice_vectorized, T_data, kappa_data, p0=initial_guess,maxfev=8_000)
#popt, pcov = curve_fit(kappa_lattice_vectorized, T_data, kappa_data, p0=initial_guess, bounds=(0, np.inf))

# Extract fitted parameters
A_fit, B_fit, C_fit = popt

# Print the fitted parameters
print(f"Fitted parameters: A = {A_fit}, B = {B_fit}, C = {C_fit}")

# Generate curve fit data
T_fit = np.linspace(min(T_data), max(T_data), 100)
kappa_fit = kappa_lattice_vectorized(T_fit, *popt)

# Plot the original data, fitted data, and extrapolated data
plt.figure(figsize=(8, 6))
plt.plot(T_data, kappa_data, 'o', label='Experimental data')
plt.plot(T_fit, kappa_fit, '-', label='Fitted model')
#plt.plot(T_extrap, kappa_extrap, '--', label='Extrapolated (400-600K)')
plt.xlabel('Temperature (K)')
plt.ylabel('Thermal Conductivity (W/mK)')
plt.legend()
plt.title('Thermal Conductivity Fitting ')
plt.show()

I suspect that the issue might be related either to the optimizer settling in a local minimum or to the lack of physical constraints on the parameters (which can allow negative values). One suggested fix is to add bounds (e.g., bounds=(0, np.inf)) in the curve_fit function to enforce nonnegative values.

Any insights or further suggestions to help ensure the fitting converges to physically realistic parameters would be greatly appreciated!

Comments 0

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Answers 2

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2 Answers

Answer by StackOverflow TychoAI Agent • 1 month ago

Below is a discussion of some aspects that could lead to a “negative” (or nonphysical) parameter value when fitting the Debye–Callaway model, along with some suggestions for how to address these issues.

1. Parameter Bounds and Physical Constraints

In many physical models (like the Debye–Callaway model), relaxation parameters such as A, B, or any additional scattering constant should be nonnegative. If no bounds are used in the fitting routine, the optimizer may wander into regions with negative values if those regions lower the sum of squared residuals. One way to enforce the constraint is to use bounds in the curve fit:

PYTHON
popt, pcov = curve_fit(
    kappa_lattice_vectorized, T_data, kappa_data, 
    p0=initial_guess, bounds=(0, np.inf), maxfev=8000)

Using bounds ensures that parameters stay in the physically meaningful (nonnegative) region.

2. Sensitivity and Local Minima

Since the model involves an integral (which itself depends on temperature and the parameters) and the optimization landscape may be relatively flat in certain regions, the optimizer can become insensitive to the initial guesses and may settle in a local minimum. Some steps you can take:

Try Different Algorithms: If curve_fit (which uses Levenberg–Marquardt by default) struggles, you might try using global optimizers (e.g., differential evolution) to search the parameter space and then refine the solution.
Parameter Scaling: Sometimes the parameters have very different magnitudes. Rescaling the parameters (or reparameterizing the model) can help the optimizer.
Fixed Parameter Approach: If one parameter is not very sensitive, consider fixing it at a literature value and fitting only the remaining parameters. This can reduce the degrees of freedom and help the optimizer find a more robust minimum.
Perform Sensitivity Analysis: Check how sensitive the fit is to each parameter by perturbing one at a time. This can sometimes reveal if a parameter is “unidentifiable,” meaning that the data are not sensitive to its value.

3. Model Implementation and Integration

Because the Debye–Callaway model includes an integral over a rapidly varying function, make sure that:

The integration limits (for example, using theta_D/T for the upper limit) are appropriate.
The integrand is computed with sufficient numerical accuracy.
You monitor the integration errors (the second return value from quad) and ensure these are small.

Sometimes, numerical issues in the integral evaluation can propagate into the parameter fitting.

4. Example Adjustments

Here’s an example snippet incorporating some of the above ideas (using bounds):

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

# Constants
kb = 1.380649e-23
hbar = 1.0545718e-34
theta_D = 280  # or update based on your case
v = 2300

def tau_PD_inv(omega, A):
    return A * omega**4

def tau_U_inv(omega, B, T):
    return B * omega**2 * np.exp(-theta_D / (3 * T))

def tau_GB_inv(omega):
    D = 0.7e-6
    return v / D

def tau_total_inv(omega, A, B, T):
    return tau_PD_inv(omega, A) + tau_U_inv(omega, B, T) + tau_GB_inv(omega)

def kappa_lattice(T, A, B):
    def integrand(x):
        omega = x * kb * T / hbar
        tau_total = 1 / tau_total_inv(omega, A, B, T)
        return x**4 * np.exp(x) / (np.exp(x) - 1)**2 * tau_total
    integral, err = quad(integrand, 0, theta_D / T)
    prefactor = kb**4 * T**3 / (2 * np.pi**2 * v * hbar**3)
    return prefactor * integral

def kappa_lattice_vectorized(T, A, B):
    return np.array([kappa_lattice(Ti, A, B) for Ti in T])

# Sample experimental data
T_data = np.array([...])  # your temperature values
kappa_data = np.array([...])  # your thermal conductivity values

# Initial guess based on literature
initial_guess = [4.75E-42, 1.054E-17]

# Use bounds to enforce non-negativity
popt, pcov = curve_fit(
    kappa_lattice_vectorized, T_data, kappa_data,
    p0=initial_guess, bounds=(0, np.inf), maxfev=8000
)

A_fit, B_fit = popt
print(f"Fitted parameters: A = {A_fit}, B = {B_fit}")

# Plot the fit
T_fit = np.linspace(min(T_data), max(T_data), 100)
kappa_fit = kappa_lattice_vectorized(T_fit, *popt)

plt.figure(figsize=(8, 6))
plt.plot(T_data, kappa_data, 'o', label='Experimental data')
plt.plot(T_fit, kappa_fit, '-', label='Fitted model')
plt.xlabel('Temperature (K)')
plt.ylabel('Thermal Conductivity (W/mK)')
plt.legend()
plt.title('Thermal Conductivity Fitting')
plt.show()

Final Thoughts

A negative fitted parameter often implies that the cost function (or least‐squares error) is minimized by an unphysical parameter value. This might be due to the optimizer getting stuck in a local minimum or to the fact that the data are insufficiently sensitive to that parameter in the given temperature range. Applying bounds, experimenting with initial guesses, or trying a global optimization scheme can help ensure that the parameters converge to physically meaningful values.

Feel free to ask if you need more specific advice on any of these points!

Tycho is an AI agent, that grounds responses in various sources like documentation, code bases, live discussions, and relevant posts. Want to chat privately with Tycho?

No comments yet.

Answer by SupernovaScientist635 • 1 month ago

Your tau_U_inv function forgot to take into account T.

The corrected version should be this:

PYTHON
def tau_U_inv(omega, B, T):
    return B * omega**2 * T * np.exp(-theta_D / (3 * T))

When I make this change the parameters show

BASH
Fitted parameters: A = 1.935907419814169e-43, B = 9.770738747258265e-18

which appears to be much closer to the original.

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Discussion

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Why Does curve_fit Return Negative Parameters in the Debye–Callaway Thermal Conductivity Model?

2 Answers

1. Parameter Bounds and Physical Constraints

2. Sensitivity and Local Minima

3. Model Implementation and Integration

4. Example Adjustments

Final Thoughts

Discussion

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