GP 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 division, unicode_literals

from numpy import array

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

configure_logger()

Out:

<RootLogger root (INFO)>

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:

    INFO - 09:23:16:
    INFO - 09:23:16: *** Start DOE Scenario execution ***
    INFO - 09:23:16: DOEScenario
    INFO - 09:23:16:    Disciplines: func
    INFO - 09:23:16:    MDOFormulation: DisciplinaryOpt
    INFO - 09:23:16:    Algorithm: fullfact
    INFO - 09:23:16: Optimization problem:
    INFO - 09:23:16:    Minimize: y_1(x_1, x_2)
    INFO - 09:23:16:    With respect to: x_1, x_2
    INFO - 09:23:16: Full factorial design required. Number of samples along each direction for a design vector of size 2 with 9 samples: 3
    INFO - 09:23:16: Final number of samples for DOE = 9 vs 9 requested
    INFO - 09:23:16: DOE sampling:   0%|          | 0/9 [00:00<?, ?it]
    INFO - 09:23:16: DOE sampling: 100%|██████████| 9/9 [00:00<00:00, 413.94 it/sec, obj=6]
    INFO - 09:23:16: Optimization result:
    INFO - 09:23:16: Objective value = 1.0
    INFO - 09:23:16: The result is feasible.
    INFO - 09:23:16: Status: None
    INFO - 09:23:16: Optimizer message: None
    INFO - 09:23:16: Number of calls to the objective function by the optimizer: 9
    INFO - 09:23:16: Design Space:
    INFO - 09:23:16: +------+-------------+-------+-------------+-------+
    INFO - 09:23:16: | name | lower_bound | value | upper_bound | type  |
    INFO - 09:23:16: +------+-------------+-------+-------------+-------+
    INFO - 09:23:16: | x_1  |      0      |   0   |      1      | float |
    INFO - 09:23:16: | x_2  |      0      |   0   |      1      | float |
    INFO - 09:23:16: +------+-------------+-------+-------------+-------+
    INFO - 09:23:16: *** DOE Scenario run terminated ***

{'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.

dataset = discipline.cache.export_to_dataset()
model = create_regression_model("GaussianProcessRegression", data=dataset)
model.learn()
print(model)

Out:

GaussianProcessRegression(alpha=1e-10, kernel='Matern', n_restarts_optimizer=10, optimizer='fmin_l_bfgs_b', random_state=None)
   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)
print(output_value)

Out:

{'y_1': array([8.50166031]), 'y_2': array([-8.56035165])}