PCE regression

We want to approximate a discipline with two inputs and two outputs:

• \(y_1=1+2x_1+3x_2\)

• \(y_2=-1-2x_1-3x_2\)

over the unit hypercube \([0,1]\times[0,1]\).

from __future__ import absolute_import, division, print_function, unicode_literals

from future import standard_library
from numpy import array

from gemseo.api import (
    configure_logger,
    create_design_space,
    create_discipline,
    create_parameter_space,
    create_scenario,
)
from gemseo.mlearning.api import create_regression_model, import_regression_model

configure_logger()

standard_library.install_aliases()

Create the discipline to learn

We can implement this analytic discipline by means of the AnalyticDiscipline class.

expressions_dict = {"y_1": "1+2*x_1+3*x_2", "y_2": "-1-2*x_1-3*x_2"}
discipline = create_discipline(
    "AnalyticDiscipline", name="func", expressions_dict=expressions_dict
)

Create the input sampling space

We create the input sampling space by adding the variables one by one.

design_space = create_design_space()
design_space.add_variable("x_1", l_b=0.0, u_b=1.0)
design_space.add_variable("x_2", l_b=0.0, u_b=1.0)

Create the learning set

We can build a learning set by means of a DOEScenario with a full factorial design of experiments. The number of samples can be equal to 9 for example.

discipline.set_cache_policy(discipline.MEMORY_FULL_CACHE)
scenario = create_scenario(
    [discipline], "DisciplinaryOpt", "y_1", design_space, scenario_type="DOE"
)
scenario.execute({"algo": "fullfact", "n_samples": 9})

Out:

{'eval_jac': False, 'algo': 'fullfact', 'n_samples': 9}

Create the regression model

Then, we build the linear regression model from the discipline cache and displays this model.

prob_space = create_parameter_space()
prob_space.add_random_variable("x_1", "OTUniformDistribution")
prob_space.add_random_variable("x_2", "OTUniformDistribution")
dataset = discipline.cache.export_to_dataset()
model = create_regression_model(
    "PCERegression", data=dataset, probability_space=prob_space, transformer=None
)
model.learn()
print(model)

Out:

PCERegression(probability_space=+---------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                  Parameter space                                                                  |
+------+-------------+-------+-------------+-------+-------------------------------+----------------+---------+------+--------------------+---------+
| name | lower_bound | value | upper_bound | type  |      Initial distribution     | Transformation | Support | Mean | Standard deviation |  Range  |
+------+-------------+-------+-------------+-------+-------------------------------+----------------+---------+------+--------------------+---------+
| x_1  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |      x_1       | [0. 1.] | 0.5  |        0.29        | [0. 1.] |
| x_2  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |      x_2       | [0. 1.] | 0.5  |        0.29        | [0. 1.] |
+------+-------------+-------+-------------+-------+-------------------------------+----------------+---------+------+--------------------+---------+, strategy=LS, degree=2, n_quad=None, stieltjes=True, sparse_param=None)
| based on the openturns library
| built from 9 learning samples

Predict output

Once it is built, we can use it for prediction.

input_value = {"x_1": array([1.0]), "x_2": array([2.0])}
output_value = model.predict(input_value)
print(output_value)

Out:

{'y_1': array([9.]), 'y_2': array([-9.])}

Save the regression model

Lastly, we save the model.

directory = model.save()

Load the regression model

In an other study, we could load this model.

loaded_model = import_regression_model(directory)
print(loaded_model)

Out:

PCERegression(probability_space=+---------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                  Parameter space                                                                  |
+------+-------------+-------+-------------+-------+-------------------------------+----------------+---------+------+--------------------+---------+
| name | lower_bound | value | upper_bound | type  |      Initial distribution     | Transformation | Support | Mean | Standard deviation |  Range  |
+------+-------------+-------+-------------+-------+-------------------------------+----------------+---------+------+--------------------+---------+
| x_1  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |      x_1       | [0. 1.] | 0.5  |        0.29        | [0. 1.] |
| x_2  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |      x_2       | [0. 1.] | 0.5  |        0.29        | [0. 1.] |
+------+-------------+-------+-------------+-------+-------------------------------+----------------+---------+------+--------------------+---------+, strategy=LS, degree=2, n_quad=None, stieltjes=True, sparse_param=None)
| based on the openturns library
| built from 0 learning samples

Use the loaded regression model

And use it!

print(loaded_model.predict(input_value))

Out:

{'y_1': array([9.]), 'y_2': array([-9.])}