RBF 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()


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 - 21:50:38:
INFO - 21:50:38: *** Start DOE Scenario execution ***
INFO - 21:50:38: DOEScenario
INFO - 21:50:38:    Disciplines: func
INFO - 21:50:38:    MDOFormulation: DisciplinaryOpt
INFO - 21:50:38:    Algorithm: fullfact
INFO - 21:50:38: Optimization problem:
INFO - 21:50:38:    Minimize: y_1(x_1, x_2)
INFO - 21:50:38:    With respect to: x_1, x_2
INFO - 21:50:38: Full factorial design required. Number of samples along each direction for a design vector of size 2 with 9 samples: 3
INFO - 21:50:38: Final number of samples for DOE = 9 vs 9 requested
INFO - 21:50:38: DOE sampling:   0%|          | 0/9 [00:00<?, ?it]
INFO - 21:50:38: DOE sampling: 100%|██████████| 9/9 [00:00<00:00, 570.77 it/sec, obj=6]
INFO - 21:50:38: Optimization result:
INFO - 21:50:38: Objective value = 1.0
INFO - 21:50:38: The result is feasible.
INFO - 21:50:38: Status: None
INFO - 21:50:38: Optimizer message: None
INFO - 21:50:38: Number of calls to the objective function by the optimizer: 9
INFO - 21:50:38: Design space:
INFO - 21:50:38: +------+-------------+-------+-------------+-------+
INFO - 21:50:38: | name | lower_bound | value | upper_bound | type  |
INFO - 21:50:38: +------+-------------+-------+-------------+-------+
INFO - 21:50:38: | x_1  |      0      |   0   |      1      | float |
INFO - 21:50:38: | x_2  |      0      |   0   |      1      | float |
INFO - 21:50:38: +------+-------------+-------+-------------+-------+
INFO - 21:50:38: *** 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("RBFRegression", data=dataset)
model.learn()
print(model)


Out:

RBFRegression(epsilon=None, function='multiquadric', norm='euclidean', smooth=0.0)
based on the scipy 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([6.45029404]), 'y_2': array([-6.45029404])}


Total running time of the script: ( 0 minutes 0.075 seconds)

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