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 numpy import array

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

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 - 13:55:47:
    INFO - 13:55:47: *** Start DOEScenario execution ***
    INFO - 13:55:47: DOEScenario
    INFO - 13:55:47:    Disciplines: func
    INFO - 13:55:47:    MDO formulation: DisciplinaryOpt
    INFO - 13:55:47: Optimization problem:
    INFO - 13:55:47:    minimize y_1(x_1, x_2)
    INFO - 13:55:47:    with respect to x_1, x_2
    INFO - 13:55:47:    over the design space:
    INFO - 13:55:47:       +------+-------------+-------+-------------+-------+
    INFO - 13:55:47:       | Name | Lower bound | Value | Upper bound | Type  |
    INFO - 13:55:47:       +------+-------------+-------+-------------+-------+
    INFO - 13:55:47:       | x_1  |      0      |  None |      1      | float |
    INFO - 13:55:47:       | x_2  |      0      |  None |      1      | float |
    INFO - 13:55:47:       +------+-------------+-------+-------------+-------+
    INFO - 13:55:47: Solving optimization problem with algorithm fullfact:
    INFO - 13:55:47:     11%|█         | 1/9 [00:00<00:00, 353.12 it/sec, obj=1]
    INFO - 13:55:47:     22%|██▏       | 2/9 [00:00<00:00, 559.61 it/sec, obj=2]
    INFO - 13:55:47:     33%|███▎      | 3/9 [00:00<00:00, 713.36 it/sec, obj=3]
    INFO - 13:55:47:     44%|████▍     | 4/9 [00:00<00:00, 828.79 it/sec, obj=2.5]
    INFO - 13:55:47:     56%|█████▌    | 5/9 [00:00<00:00, 918.11 it/sec, obj=3.5]
    INFO - 13:55:47:     67%|██████▋   | 6/9 [00:00<00:00, 990.00 it/sec, obj=4.5]
    INFO - 13:55:47:     78%|███████▊  | 7/9 [00:00<00:00, 1038.30 it/sec, obj=4]
    INFO - 13:55:47:     89%|████████▉ | 8/9 [00:00<00:00, 1085.52 it/sec, obj=5]
    INFO - 13:55:47:    100%|██████████| 9/9 [00:00<00:00, 1127.00 it/sec, obj=6]
    INFO - 13:55:47: Optimization result:
    INFO - 13:55:47:    Optimizer info:
    INFO - 13:55:47:       Status: None
    INFO - 13:55:47:       Message: None
    INFO - 13:55:47:       Number of calls to the objective function by the optimizer: 9
    INFO - 13:55:47:    Solution:
    INFO - 13:55:47:       Objective: 1.0
    INFO - 13:55:47:       Design space:
    INFO - 13:55:47:          +------+-------------+-------+-------------+-------+
    INFO - 13:55:47:          | Name | Lower bound | Value | Upper bound | Type  |
    INFO - 13:55:47:          +------+-------------+-------+-------------+-------+
    INFO - 13:55:47:          | x_1  |      0      |   0   |      1      | float |
    INFO - 13:55:47:          | x_2  |      0      |   0   |      1      | float |
    INFO - 13:55:47:          +------+-------------+-------+-------------+-------+
    INFO - 13:55:47: *** End DOEScenario execution (time: 0:00:00.019796) ***

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

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.to_dataset(opt_naming=False)
model = create_regression_model(
    "PCERegressor", data=dataset, probability_space=prob_space, transformer=None
)
model.learn()
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=Uncertain space: +------+-------------------------------+ | Name | Distribution | +------+-------------------------------+ | x_1 | Uniform(lower=0.0, upper=1.0) | | x_2 | 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)
output_value

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

Save the regression model

Lastly, we save the model.

directory = model.to_pickle()

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/stable/lib/python3.9/site-packages/gemseo/mlearning/core/ml_algo.py:359: UserWarning: Pandas doesn't allow columns to be created via a new attribute name - see https://pandas.pydata.org/pandas-docs/stable/indexing.html#attribute-access
  self.learning_set.data = {}

Load the regression model

In an other study, we could load this model.

loaded_model = import_regression_model(directory)
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=Uncertain space: +------+-------------------------------+ | Name | Distribution | +------+-------------------------------+ | x_1 | Uniform(lower=0.0, upper=1.0) | | x_2 | Uniform(lower=0.0, upper=1.0) | +------+-------------------------------+, use_cleaning=False, use_lars=False)

• based on the OpenTURNS library
• built from 9 learning samples

Use the loaded regression model

And use it!

loaded_model.predict(input_value)

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