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

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", lower_bound=0.0, upper_bound=1.0)
design_space.add_variable("x_2", lower_bound=0.0, upper_bound=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],
    "y_1",
    design_space,
    scenario_type="DOE",
    formulation_name="DisciplinaryOpt",
)
scenario.execute(algo_name="PYDOE_FULLFACT", n_samples=9)

INFO - 08:37:49:
INFO - 08:37:49: *** Start DOEScenario execution ***
INFO - 08:37:49: DOEScenario
INFO - 08:37:49:    Disciplines: func
INFO - 08:37:49:    MDO formulation: DisciplinaryOpt
INFO - 08:37:49: Optimization problem:
INFO - 08:37:49:    minimize y_1(x_1, x_2)
INFO - 08:37:49:    with respect to x_1, x_2
INFO - 08:37:49:    over the design space:
INFO - 08:37:49:       +------+-------------+-------+-------------+-------+
INFO - 08:37:49:       | Name | Lower bound | Value | Upper bound | Type  |
INFO - 08:37:49:       +------+-------------+-------+-------------+-------+
INFO - 08:37:49:       | x_1  |      0      |  None |      1      | float |
INFO - 08:37:49:       | x_2  |      0      |  None |      1      | float |
INFO - 08:37:49:       +------+-------------+-------+-------------+-------+
INFO - 08:37:49: Solving optimization problem with algorithm PYDOE_FULLFACT:
INFO - 08:37:49:     11%|█         | 1/9 [00:00<00:00, 397.00 it/sec, obj=1]
INFO - 08:37:49:     22%|██▏       | 2/9 [00:00<00:00, 659.43 it/sec, obj=2]
INFO - 08:37:49:     33%|███▎      | 3/9 [00:00<00:00, 865.10 it/sec, obj=3]
INFO - 08:37:49:     44%|████▍     | 4/9 [00:00<00:00, 1028.08 it/sec, obj=2.5]
INFO - 08:37:49:     56%|█████▌    | 5/9 [00:00<00:00, 1154.37 it/sec, obj=3.5]
INFO - 08:37:49:     67%|██████▋   | 6/9 [00:00<00:00, 1263.28 it/sec, obj=4.5]
INFO - 08:37:49:     78%|███████▊  | 7/9 [00:00<00:00, 1349.15 it/sec, obj=4]
INFO - 08:37:49:     89%|████████▉ | 8/9 [00:00<00:00, 1416.76 it/sec, obj=5]
INFO - 08:37:49:    100%|██████████| 9/9 [00:00<00:00, 1482.55 it/sec, obj=6]
INFO - 08:37:49: Optimization result:
INFO - 08:37:49:    Optimizer info:
INFO - 08:37:49:       Status: None
INFO - 08:37:49:       Message: None
INFO - 08:37:49:       Number of calls to the objective function by the optimizer: 9
INFO - 08:37:49:    Solution:
INFO - 08:37:49:       Objective: 1.0
INFO - 08:37:49:       Design space:
INFO - 08:37:49:          +------+-------------+-------+-------------+-------+
INFO - 08:37:49:          | Name | Lower bound | Value | Upper bound | Type  |
INFO - 08:37:49:          +------+-------------+-------+-------------+-------+
INFO - 08:37:49:          | x_1  |      0      |   0   |      1      | float |
INFO - 08:37:49:          | x_2  |      0      |   0   |      1      | float |
INFO - 08:37:49:          +------+-------------+-------+-------------+-------+
INFO - 08:37:49: *** End DOEScenario execution (time: 0:00:00.010095) ***

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", dataset, probability_space=prob_space, transformer={}
)
model.learn()
model

PCERegressor(cleaning_options=None, degree=2, discipline=None, hyperbolic_parameter=1.0, input_names=(), n_quadrature_points=0, output_names=(), parameters={}, probability_space=Uncertain space: +------+-------------------------------+ | Name | Distribution | +------+-------------------------------+ | x_1 | Uniform(lower=0.0, upper=1.0) | | x_2 | Uniform(lower=0.0, upper=1.0) | +------+-------------------------------+, transformer={}, use_cleaning=False, use_lars=False, use_quadrature=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.])}