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 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", "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 - 13:27:07:
INFO - 13:27:07: *** Start DOEScenario execution ***
INFO - 13:27:07: DOEScenario
INFO - 13:27:07:    Disciplines: func
INFO - 13:27:07:    MDO formulation: DisciplinaryOpt
INFO - 13:27:07: Optimization problem:
INFO - 13:27:07:    minimize y_1(x_1, x_2)
INFO - 13:27:07:    with respect to x_1, x_2
INFO - 13:27:07:    over the design space:
INFO - 13:27:07:       +------+-------------+-------+-------------+-------+
INFO - 13:27:07:       | Name | Lower bound | Value | Upper bound | Type  |
INFO - 13:27:07:       +------+-------------+-------+-------------+-------+
INFO - 13:27:07:       | x_1  |      0      |  None |      1      | float |
INFO - 13:27:07:       | x_2  |      0      |  None |      1      | float |
INFO - 13:27:07:       +------+-------------+-------+-------------+-------+
INFO - 13:27:07: Solving optimization problem with algorithm PYDOE_FULLFACT:
INFO - 13:27:07:     11%|█         | 1/9 [00:00<00:00, 361.77 it/sec, obj=1]
INFO - 13:27:07:     22%|██▏       | 2/9 [00:00<00:00, 612.58 it/sec, obj=2]
INFO - 13:27:07:     33%|███▎      | 3/9 [00:00<00:00, 810.18 it/sec, obj=3]
INFO - 13:27:07:     44%|████▍     | 4/9 [00:00<00:00, 971.18 it/sec, obj=2.5]
INFO - 13:27:07:     56%|█████▌    | 5/9 [00:00<00:00, 1103.42 it/sec, obj=3.5]
INFO - 13:27:07:     67%|██████▋   | 6/9 [00:00<00:00, 1214.16 it/sec, obj=4.5]
INFO - 13:27:07:     78%|███████▊  | 7/9 [00:00<00:00, 1305.18 it/sec, obj=4]
INFO - 13:27:07:     89%|████████▉ | 8/9 [00:00<00:00, 1384.54 it/sec, obj=5]
INFO - 13:27:07:    100%|██████████| 9/9 [00:00<00:00, 1447.53 it/sec, obj=6]
INFO - 13:27:07: Optimization result:
INFO - 13:27:07:    Optimizer info:
INFO - 13:27:07:       Status: None
INFO - 13:27:07:       Message: None
INFO - 13:27:07:       Number of calls to the objective function by the optimizer: 9
INFO - 13:27:07:    Solution:
INFO - 13:27:07:       Objective: 1.0
INFO - 13:27:07:       Design space:
INFO - 13:27:07:          +------+-------------+-------+-------------+-------+
INFO - 13:27:07:          | Name | Lower bound | Value | Upper bound | Type  |
INFO - 13:27:07:          +------+-------------+-------+-------------+-------+
INFO - 13:27:07:          | x_1  |      0      |   0   |      1      | float |
INFO - 13:27:07:          | x_2  |      0      |   0   |      1      | float |
INFO - 13:27:07:          +------+-------------+-------+-------------+-------+
INFO - 13:27:07: *** End DOEScenario execution (time: 0:00:00.010377) ***

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("GaussianProcessRegressor", data=dataset)
model.learn()
model
GaussianProcessRegressor(alpha=1e-10, bounds=(), input_names=(), kernel=None, n_restarts_optimizer=10, optimizer=fmin_l_bfgs_b, output_names=(), parameters={}, random_state=0, transformer={'inputs': <gemseo.mlearning.transformers.scaler.min_max_scaler.MinMaxScaler object at 0x7fc6111992b0>, 'outputs': <gemseo.mlearning.transformers.scaler.min_max_scaler.MinMaxScaler object at 0x7fc6111994c0>})
  • 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([6.03823023])}

Total running time of the script: (0 minutes 0.095 seconds)

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