Surrogate modeling methods

SMT contains the surrogate modeling methods listed below.

• Least-squares approximation
• Second-order polynomial approximation
• Gausian process regression
  • Kriging (KRG)
  • KPLS
  • KPLSK
  • GPX
  • GEKPLS
  • Marginal Gaussian Process (MGP)
  • Sparse Gaussian Process (SGP)
• Radial basis functions
• Inverse-distance weighting
• Regularized minimal-energy tensor-product splines
• GENN

Usage

import matplotlib.pyplot as plt
import numpy as np

from smt.surrogate_models import RBF

xt = np.array([0.0, 1.0, 2.0, 3.0, 4.0])
yt = np.array([0.0, 1.0, 1.5, 0.9, 1.0])

sm = RBF(d0=5)
sm.set_training_values(xt, yt)
sm.train()

num = 100
x = np.linspace(0.0, 4.0, num)
y = sm.predict_values(x)

plt.plot(xt, yt, "o")
plt.plot(x, y)
plt.xlabel("x")
plt.ylabel("y")
plt.legend(["Training data", "Prediction"])
plt.show()

___________________________________________________________________________

                                    RBF
___________________________________________________________________________

 Problem size

      # training points.        : 5

___________________________________________________________________________

 Training

   Training ...
      Initializing linear solver ...
         Performing LU fact. (5 x 5 mtx) ...
         Performing LU fact. (5 x 5 mtx) - done. Time (sec):  0.0000000
      Initializing linear solver - done. Time (sec):  0.0000000
      Solving linear system (col. 0) ...
         Back solving (5 x 5 mtx) ...
         Back solving (5 x 5 mtx) - done. Time (sec):  0.0000000
      Solving linear system (col. 0) - done. Time (sec):  0.0000000
   Training - done. Time (sec):  0.0000000
___________________________________________________________________________

 Evaluation

      # eval points. : 100

   Predicting ...
   Predicting - done. Time (sec):  0.0000000

   Prediction time/pt. (sec) :  0.0000000

SurrogateModel class API

All surrogate modeling methods implement the following API, though some of the functions in the API are not supported by all methods.

class smt.surrogate_models.surrogate_model.SurrogateModel(**kwargs)

Base class for all surrogate models.

Attributes:

optionsOptionsDictionary

Dictionary of options. Options values can be set on this attribute directly or they can be passed in as keyword arguments during instantiation.

supportsdict

Dictionary containing information about what this surrogate model supports.

Methods

load(filename)	Static method implemented by each surrogate model type (ie class) to load the surrogate object from a file created by using the corresponding save method
predict_derivatives(x, kx)	Predict the dy_dx derivatives at a set of points.
predict_output_derivatives(x)	Predict the derivatives dy_dyt at a set of points.
predict_values(x)	Predict the output values at a set of points.
predict_variance_derivatives(x, kx)	Provide the derivatives of the variance of the model at a set of points.
predict_variance_gradient(x)	Provide the gradient of the variance of the model at a given point (ie the derivatives wrt to all component at a unique point x)
predict_variances(x)	Predict the variances at a set of points.
save(filename)	Implemented by surrogate models to save the surrogate object in a file
set_training_derivatives(xt, dyt_dxt, kx[, name])	Set training data (derivatives).
set_training_values(xt, yt[, name])	Set training data (values).
train()	Train the model
update_training_derivatives(dyt_dxt, kx[, name])	Update the training data (values) at the previously set input values.
update_training_values(yt[, name])	Update the training data (values) at the previously set input values.

Examples

>>> from smt.surrogate_models import RBF
>>> sm = RBF(print_training=False)
>>> sm.options['print_prediction'] = False

__init__(**kwargs)

Constructor where values of options can be passed in.

For the list of options, see the documentation for the surrogate model being used.

Parameters:

**kwargsnamed arguments

Set of options that can be optionally set; each option must have been declared.

Examples

>>> from smt.surrogate_models import RBF
>>> sm = RBF(print_global=False)

set_training_values(xt: ndarray, yt: ndarray, name=None) → None

Set training data (values).

Parameters:

xtnp.ndarray[nt, nx] or np.ndarray[nt]

The input values for the nt training points.

ytnp.ndarray[nt, ny] or np.ndarray[nt]

The output values for the nt training points.

namestr or None

An optional label for the group of training points being set. This is only used in special situations (e.g., multi-fidelity applications).

set_training_derivatives(xt: ndarray, dyt_dxt: ndarray, kx: int, name: str | None = None) → None

Set training data (derivatives).

Parameters:

xtnp.ndarray[nt, nx] or np.ndarray[nt]

The input values for the nt training points.

dyt_dxtnp.ndarray[nt, ny] or np.ndarray[nt]

The derivatives values for the nt training points.

kxint

0-based index of the derivatives being set.

namestr or None

An optional label for the group of training points being set. This is only used in special situations (e.g., multi-fidelity applications).

train() → None

Train the model

predict_values(x: ndarray) → ndarray

Predict the output values at a set of points.

Parameters:

xnp.ndarray[nt, nx] or np.ndarray[nt]

Input values for the prediction points.

Returns:

ynp.ndarray[nt, ny]

Output values at the prediction points.

predict_derivatives(x: ndarray, kx: int) → ndarray

Predict the dy_dx derivatives at a set of points.

Parameters:

xnp.ndarray[nt, nx] or np.ndarray[nt]

Input values for the prediction points.

kxint

The 0-based index of the input variable with respect to which derivatives are desired.

Returns:

dy_dxnp.ndarray[nt, ny]

Derivatives.

predict_output_derivatives(x: ndarray) → dict

Predict the derivatives dy_dyt at a set of points.

Parameters:

xnp.ndarray[nt, nx] or np.ndarray[nt]

Input values for the prediction points.

Returns:

dy_dytdict of np.ndarray[nt, nt]

Dictionary of output derivatives. Key is None for derivatives wrt yt and kx for derivatives wrt dyt_dxt.

predict_variances(x: ndarray) → ndarray

Predict the variances at a set of points.

Parameters:

xnp.ndarray[nt, nx] or np.ndarray[nt]

Input values for the prediction points.

Returns:

s2np.ndarray[nt, ny]

Variances.

predict_variance_derivatives(x: ndarray, kx: int) → ndarray

Provide the derivatives of the variance of the model at a set of points.

Parameters:

xnp.ndarray [n_evals, dim]

Evaluation point input variable values

kxint

The 0-based index of the input variable with respect to which derivatives are desired.

Returns:

derived_variance: np.ndarray

The kx-th derivatives of the variance of the kriging model

predict_variance_gradient(x: ndarray) → ndarray

Provide the gradient of the variance of the model at a given point (ie the derivatives wrt to all component at a unique point x)

Parameters:

xnp.ndarray [1, dim] or even (dim,) vector

Evaluation point input variable values

Returns:

derived_variancenp.ndarray

The jacobian of the variance of the kriging model

save(filename)

Implemented by surrogate models to save the surrogate object in a file

static load(filename)

Static method implemented by each surrogate model type (ie class) to load the surrogate object from a file created by using the corresponding save method

How to save and load trained surrogate models

As of SMT 2.9, the surrogate API offers save/load methods which can be used as below.

Saving the model

The instance method save() is used to save the trained model (here KRG) in a binary file named kriging.bin.

sm = KRG()
sm.set_training_values(xtrain, ytrain)
sm.train()

sm.save("kriging.bin ")

Loading the model

The previous model can be reloaded knowing the type of surrogate being saved (ie. KRG) from the binary file with:

sm2 = KRG.load("kriging.bin")
ytest = sm2.predict_values(xtest)