Polynomial 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_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 + x_1**2",
    "y_2": "-1 - 2*x_1 + x_1*x_2 - 3*x_2**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 - 09:01:51:
    INFO - 09:01:51: *** Start DOEScenario execution ***
    INFO - 09:01:51: DOEScenario
    INFO - 09:01:51:    Disciplines: func
    INFO - 09:01:51:    MDO formulation: DisciplinaryOpt
    INFO - 09:01:51: Optimization problem:
    INFO - 09:01:51:    minimize y_1(x_1, x_2)
    INFO - 09:01:51:    with respect to x_1, x_2
    INFO - 09:01:51:    over the design space:
    INFO - 09:01:51:       +------+-------------+-------+-------------+-------+
    INFO - 09:01:51:       | Name | Lower bound | Value | Upper bound | Type  |
    INFO - 09:01:51:       +------+-------------+-------+-------------+-------+
    INFO - 09:01:51:       | x_1  |      0      |  None |      1      | float |
    INFO - 09:01:51:       | x_2  |      0      |  None |      1      | float |
    INFO - 09:01:51:       +------+-------------+-------+-------------+-------+
    INFO - 09:01:51: Solving optimization problem with algorithm fullfact:
    INFO - 09:01:51:     11%|█         | 1/9 [00:00<00:00, 289.86 it/sec, obj=1]
    INFO - 09:01:51:     22%|██▏       | 2/9 [00:00<00:00, 467.54 it/sec, obj=2.25]
    INFO - 09:01:51:     33%|███▎      | 3/9 [00:00<00:00, 608.87 it/sec, obj=4]
    INFO - 09:01:51:     44%|████▍     | 4/9 [00:00<00:00, 721.10 it/sec, obj=2.5]
    INFO - 09:01:51:     56%|█████▌    | 5/9 [00:00<00:00, 808.74 it/sec, obj=3.75]
    INFO - 09:01:51:     67%|██████▋   | 6/9 [00:00<00:00, 882.52 it/sec, obj=5.5]
    INFO - 09:01:51:     78%|███████▊  | 7/9 [00:00<00:00, 939.49 it/sec, obj=4]
    INFO - 09:01:51:     89%|████████▉ | 8/9 [00:00<00:00, 990.30 it/sec, obj=5.25]
    INFO - 09:01:51:    100%|██████████| 9/9 [00:00<00:00, 1035.20 it/sec, obj=7]
    INFO - 09:01:51: Optimization result:
    INFO - 09:01:51:    Optimizer info:
    INFO - 09:01:51:       Status: None
    INFO - 09:01:51:       Message: None
    INFO - 09:01:51:       Number of calls to the objective function by the optimizer: 9
    INFO - 09:01:51:    Solution:
    INFO - 09:01:51:       Objective: 1.0
    INFO - 09:01:51:       Design space:
    INFO - 09:01:51:          +------+-------------+-------+-------------+-------+
    INFO - 09:01:51:          | Name | Lower bound | Value | Upper bound | Type  |
    INFO - 09:01:51:          +------+-------------+-------+-------------+-------+
    INFO - 09:01:51:          | x_1  |      0      |   0   |      1      | float |
    INFO - 09:01:51:          | x_2  |      0      |   0   |      1      | float |
    INFO - 09:01:51:          +------+-------------+-------+-------------+-------+
    INFO - 09:01:51: *** End DOEScenario execution (time: 0:00:00.020725) ***

{'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.

dataset = scenario.to_dataset(opt_naming=False)
model = create_regression_model(
    "PolynomialRegressor", data=dataset, degree=2, transformer=None
)
model.learn()
model

PolynomialRegressor(degree=2, fit_intercept=True, l2_penalty_ratio=1.0, penalty_level=0.0, random_state=0)

• based on the scikit-learn 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([10.])}

Predict Jacobian

We can also use it to predict the jacobian of the discipline.

jacobian_value = model.predict_jacobian(input_value)
jacobian_value

{'y_1': {'x_1': array([[4.]]), 'x_2': array([[3.]])}}

Get intercept

In addition, it is possible to access the intercept of the model, either directly or by means of a method returning either a dictionary (default option) or an array.

model.intercept, model.get_intercept()

(array([1.]), {'y_1': [1.0]})

Get coefficients

In addition, it is possible to access the coefficients of the model, either directly or by means of a method returning either a dictionary (default option) or an array.

model.coefficients

array([[ 2.00000000e+00,  3.00000000e+00,  1.00000000e+00,
        -6.22449391e-16, -5.07973866e-16]])