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 annotations

from gemseo.api import configure_logger
from gemseo.api import create_design_space
from gemseo.api import create_discipline
from gemseo.api import create_parameter_space
from gemseo.api import create_scenario
from gemseo.mlearning.api import create_regression_model
from gemseo.mlearning.api import import_regression_model
from numpy import array

configure_logger()
<RootLogger root (INFO)>

Create the discipline to learn

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

expressions = {"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=expressions
)

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.

scenario = create_scenario(
    [discipline], "DisciplinaryOpt", "y_1", design_space, scenario_type="DOE"
)
scenario.execute({"algo": "fullfact", "n_samples": 9})
    INFO - 11:24:53:
    INFO - 11:24:53: *** Start DOEScenario execution ***
    INFO - 11:24:53: DOEScenario
    INFO - 11:24:53:    Disciplines: func
    INFO - 11:24:53:    MDO formulation: DisciplinaryOpt
    INFO - 11:24:53: Optimization problem:
    INFO - 11:24:53:    minimize y_1(x_1, x_2)
    INFO - 11:24:53:    with respect to x_1, x_2
    INFO - 11:24:53:    over the design space:
    INFO - 11:24:53:    +------+-------------+-------+-------------+-------+
    INFO - 11:24:53:    | name | lower_bound | value | upper_bound | type  |
    INFO - 11:24:53:    +------+-------------+-------+-------------+-------+
    INFO - 11:24:53:    | x_1  |      0      |  None |      1      | float |
    INFO - 11:24:53:    | x_2  |      0      |  None |      1      | float |
    INFO - 11:24:53:    +------+-------------+-------+-------------+-------+
    INFO - 11:24:53: Solving optimization problem with algorithm fullfact:
    INFO - 11:24:53: ...   0%|          | 0/9 [00:00<?, ?it]
    INFO - 11:24:53: ... 100%|██████████| 9/9 [00:00<00:00, 1453.33 it/sec, obj=6]
    INFO - 11:24:53: Optimization result:
    INFO - 11:24:53:    Optimizer info:
    INFO - 11:24:53:       Status: None
    INFO - 11:24:53:       Message: None
    INFO - 11:24:53:       Number of calls to the objective function by the optimizer: 9
    INFO - 11:24:53:    Solution:
    INFO - 11:24:53:       Objective: 1.0
    INFO - 11:24:53:       Design space:
    INFO - 11:24:53:       +------+-------------+-------+-------------+-------+
    INFO - 11:24:53:       | name | lower_bound | value | upper_bound | type  |
    INFO - 11:24:53:       +------+-------------+-------+-------------+-------+
    INFO - 11:24:53:       | x_1  |      0      |   0   |      1      | float |
    INFO - 11:24:53:       | x_2  |      0      |   0   |      1      | float |
    INFO - 11:24:53:       +------+-------------+-------+-------------+-------+
    INFO - 11:24:53: *** End DOEScenario execution (time: 0:00:00.015358) ***

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

Create the regression model

Then, we build the linear regression model from the database 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 = scenario.export_to_dataset(opt_naming=False)
model = create_regression_model(
    "PCERegressor", data=dataset, probability_space=prob_space, transformer=None
)
model.learn()
print(model)
PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=+----------------------------------------------------------------------------------+
|                                 Parameter space                                  |
+------+-------------+-------+-------------+-------+-------------------------------+
| name | lower_bound | value | upper_bound | type  |      Initial distribution     |
+------+-------------+-------+-------------+-------+-------------------------------+
| x_1  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
| x_2  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
+------+-------------+-------+-------------+-------+-------------------------------+, use_cleaning=False, use_lars=False)
   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)
{'y_1': 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)
PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=+----------------------------------------------------------------------------------+
|                                 Parameter space                                  |
+------+-------------+-------+-------------+-------+-------------------------------+
| name | lower_bound | value | upper_bound | type  |      Initial distribution     |
+------+-------------+-------+-------------+-------+-------------------------------+
| x_1  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
| x_2  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
+------+-------------+-------+-------------+-------+-------------------------------+, use_cleaning=False, use_lars=False)
   based on the OpenTURNS library
   built from 0 learning samples

Use the loaded regression model

And use it!

print(loaded_model.predict(input_value))
{'y_1': array([9.])}

Total running time of the script: ( 0 minutes 0.174 seconds)

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