pyoptmat.temperature: dealing with temperature dependence

The idea of the objects here, derived from TemperatureParameter is to provide temperature-dependent versions of raw model parameters.

These scaling rules are functions of some “actual” parameters, either PyTorch or Pyro parameters, which are actually optimized. These scaling rules somehow map a collection of these parameters to a temperature dependent property.

On top of this temperature scaling, these classes can also, optionally be used to do numerical scaling of the underlying parameters to achieve better numerical stability. So, in the end, mathematically these classes are functions of the kind

\[v = s(t(T;p))\]

where \(s\) is a scaling function with fixed parameters aimed at helping numerical scaling and \(t\) is the actual temperature scaling function, parameterized with a set of parameters \(p\) which will be varied during optimization.

The simplest example is the pyoptmat.temperature.ConstantParameter which simple returns the constant (temperature-indpendent) value of a single underlying parameter p.

The second simplest example is pyoptmat.temperature.PolynomialScaling which takes a tensor parameter c describing polynomial coefficients. This scaling function evaluates the polynomial described by these coefficients at T to provide the parameter temperature dependence.

The numerical scaling functions \(s\) default to the identity, i.e. no scaling. The pyoptmat.optimize module provides some common, useful scaling functions, though often simply scaling by an appropriate order of magnitude value so that the “actual” parameters \(p\) are on the order of 1.

All of these temperature scaling rules have to be compatible with batched temperatures, i.e. they should expect to receive a tensor of shape (nbatch,), not just a scalar.

class pyoptmat.temperature.ArrheniusScaling(A, Q, *args, A_scale=<function ArrheniusScaling.<lambda>>, Q_scale=<function ArrheniusScaling.<lambda>>, **kwargs)

Bases: TemperatureParameter

Simple Arrhenius scaling of the type

\[A \exp(-Q/T)\]

Parameters:

A (torch.tensor) – Prefactor
Q (torch.tensor) – Activation energy (times R)

Keyword Args: A_scale (function): numerical scaling for A, defaults to no scaling Q_scale (function): numerical scaling for Q, defaults to no scaling

property device: Return the device used by the scaling function

property shape: Shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.ConstantParameter(pvalue, *args, p_scale=<function ConstantParameter.<lambda>>, **kwargs)

Bases: TemperatureParameter

A parameter that is constant with temperature

Parameters:: pvalue (torch.tensor) – the constant parameter value
Keyword Arguments:: p_scale (function) – numerical scaling function, defaults to no scaling

property device: Return the device used by the scaling function

property shape: The shape of the underlying parameter

value(T)

Pretty simple, just return the value!

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.InverseArrheniusScaling(A, Q, *args, A_scale=<function InverseArrheniusScaling.<lambda>>, Q_scale=<function InverseArrheniusScaling.<lambda>>, **kwargs)

Bases: TemperatureParameter

Simple Arrhenius scaling of the type

\[A (1 - \exp(-Q/T))\]

Parameters:

A (torch.tensor) – Prefactor
Q (torch.tensor) – Activation energy (times R)

Keyword Args: A_scale (function): numerical scaling for A, defaults to no scaling Q_scale (function): numerical scaling for Q, defaults to no scaling

property device: Return the device used by the scaling function

property shape: Shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.KMRateSensitivityScaling(A, mu, b, k, *args, A_scale=<function KMRateSensitivityScaling.<lambda>>, cutoff=20.0, **kwargs)

Bases: TemperatureParameter

Parameter that scales as:

\[\frac{-\mu b^3}{kTA}\]

where \(\mu\) further depends on temperature

Parameters:

A (torch.tensor) – Kocks-Mecking slope parameter, sets shape
mu (TemperatureParameter) – scalar, temperature-dependent shear modulus
b (torch.tensor) – scalar, Burgers vector
k (torch.tensor) – scalar, Boltzmann constant

Keyword Arguments:

cutoff (float) – don’t let n go higher than this
A_scale (function) – numerical scaling function for A, defaults to no scaling

property device: Return the device used by the scaling function

property shape: Shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.KMViscosityScaling(A, B, mu, eps0, b, k, *args, A_scale=<function KMViscosityScaling.<lambda>>, B_scale=<function KMViscosityScaling.<lambda>>, **kwargs)

Bases: TemperatureParameter

Parameter that varies as

\[\exp{B} \mu \dot{\varepsilon}_0^{-1/n}\]

where \(B\) is the Kocks-Mecking intercept parameter and the rest are defined in the pyoptmat.temperature.KMRateSensitivityScaling object.

\(n\) is the rate sensitivity, again given by the pyoptmat.temperature.KMRateSensitivityScaling object

Parameters:

A (torch.tensor) – Kocks-Mecking slope parameter
B (torch.tensor) – Kocks-Mecking intercept parameter, sets shape, must be same shape as A
mu (TemperatureParameter) – scalar, temperature-dependent shear modulus
eps0 (torch.tensor) – scalar, reference strain rate
b (torch.tensor) – scalar, Burger’s vector
k (torch.tensor) – scalar, Boltzmann constant

Keyword Arguments:

A_scale (function) – numerical scaling function for A, defaults to no scaling
B_scale (function) – numerical scaling function B, defaults to no scaling

property device: Return the device used by the scaling function

property shape: Shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.MTSScaling(tau0, g0, q, p, k, b, mu, *args, tau0_scale=<function MTSScaling.<lambda>>, g0_scale=<function MTSScaling.<lambda>>, q_scale=<function MTSScaling.<lambda>>, p_scale=<function MTSScaling.<lambda>>, **kwargs)

Bases: TemperatureParameter

Parameter that scales as:

\[\hat{\tau}\left\{ 1 - \left[ \frac{kT}{\mu b^3 g_0} \right]^{1/q} \right\}^{1/p}\]

Parameters:

tau0 (torch.tensor) – threshold strength
g0 (torch.tensor) – activation energy
q (torch.tensor) – shape parameter
p (torch.tensor) – shape parameter
k (torch.tensor) – Boltzmann constant
b (torch.tensor) – burgers vector
mu (TemperatureParameter) – shear modulus, temperature-dependent

Keyword Arguments:

tau0_scale (function) – numerical scaling function for tau0, defaults to no scaling
g0_scale (function) – numerical scaling function for g0, defaults to no scaling
q_scale (function) – numerical scaling function for g0, defaults to no scaling
p_scale (function) – numerical scaling function for p, defaults to no scaling

property device: Return the device used by the scaling function

property shape: Shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.PiecewiseScaling(control, values, *args, values_scale_fn=<function PiecewiseScaling.<lambda>>, **kwargs)

Bases: TemperatureParameter

Piecewise linear interpolation, mimics scipy.interp1d with default options

Parameters:

control (torch.tensor) – temperature control points (npoints)
values (torch.tensor) – values at control points (optional_batch, npoints)

Keyword Arguments:

values_scale_fn (function) – numerical scaling function for values, defaults to no scaling

property device: Return the device used by the scaling function

property shape: Shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.PolynomialScaling(coefs, *args, coef_scale_fn=<function PolynomialScaling.<lambda>>, **kwargs)

Bases: TemperatureParameter

Mimics np.polyval using Horner’s method to evaluate a polynomial

Parameters:: coefs (torch.tensor) – polynomial coefficients in the numpy convention (highest order 1st)
Keyword Arguments:: coef_scale_fn (function) – numerical scaling function for coefs, defaults to no scaling

property device: Return the device used by the scaling function

property shape: Shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.ShearModulusScaling(A, mu, *args, A_scale=<function ShearModulusScaling.<lambda>>, **kwargs)

Bases: TemperatureParameter

Parameter that scales as:

\[A \mu\]

where \(\mu\) further depends on temperature

Parameters:

A (torch.tensor) – actual parameter
mu (TemperatureParameter) – scalar, temperature-dependent shear modulus

Keyword Arguments:

A_scale (function) – numerical scaling function for A, defaults to no scaling

property device: Return the device used by the scaling function

property shape: The shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.ShearModulusScalingExp(A, mu, *args, A_scale=<function ShearModulusScalingExp.<lambda>>, **kwargs)

Bases: TemperatureParameter

Parameter that scales as:

\[\exp(A) \mu\]

where \(\mu\) further depends on temperature

Parameters:

A (torch.tensor) – actual parameter
mu (TemperatureParameter) – scalar, temperature-dependent shear modulus

Keyword Arguments:

A_scale (function) – numerical scaling function for A, defaults to no scaling

property device: Return the device used by the scaling function

property shape: The shape of the underlying parameter

value(T)

Return the function value

Parameters:: T (torch.tensor) – current temperature
Returns:: value at the given temperatures
Return type:: torch.tensor

class pyoptmat.temperature.TemperatureParameter(*args, scaling=<function TemperatureParameter.<lambda>>, **kwargs)

Bases: Module

Superclass of all temperature-dependent parameters

This class takes care of numerical scaling on the end result, if required

Keyword Arguments:: scaling (function) – numerical scaling function, defaults to no scaling

forward(T)

Return the actual parameter value

Parameters:: T (torch.tensor) – current temperature