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 annotations

from gemseo.api import configure_logger
from gemseo.api import create_design_space
from gemseo.api import create_discipline
from gemseo.api import create_scenario
from gemseo.mlearning.api import create_regression_model
from numpy import array

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", 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 - 17:25:59:
    INFO - 17:25:59: *** Start DOEScenario execution ***
    INFO - 17:25:59: DOEScenario
    INFO - 17:25:59:    Disciplines: func
    INFO - 17:25:59:    MDO formulation: DisciplinaryOpt
    INFO - 17:25:59: Optimization problem:
    INFO - 17:25:59:    minimize y_1(x_1, x_2)
    INFO - 17:25:59:    with respect to x_1, x_2
    INFO - 17:25:59:    over the design space:
    INFO - 17:25:59:    +------+-------------+-------+-------------+-------+
    INFO - 17:25:59:    | name | lower_bound | value | upper_bound | type  |
    INFO - 17:25:59:    +------+-------------+-------+-------------+-------+
    INFO - 17:25:59:    | x_1  |      0      |  None |      1      | float |
    INFO - 17:25:59:    | x_2  |      0      |  None |      1      | float |
    INFO - 17:25:59:    +------+-------------+-------+-------------+-------+
    INFO - 17:25:59: Solving optimization problem with algorithm fullfact:
    INFO - 17:25:59: ...   0%|          | 0/9 [00:00<?, ?it]
    INFO - 17:25:59: ...  11%|█         | 1/9 [00:00<00:00, 204.71 it/sec, obj=1]
    INFO - 17:25:59: ...  22%|██▏       | 2/9 [00:00<00:00, 337.91 it/sec, obj=2]
    INFO - 17:25:59: ...  33%|███▎      | 3/9 [00:00<00:00, 430.58 it/sec, obj=3]
    INFO - 17:25:59: ...  44%|████▍     | 4/9 [00:00<00:00, 500.75 it/sec, obj=2.5]
    INFO - 17:25:59: ...  56%|█████▌    | 5/9 [00:00<00:00, 557.74 it/sec, obj=3.5]
    INFO - 17:25:59: ...  67%|██████▋   | 6/9 [00:00<00:00, 603.58 it/sec, obj=4.5]
    INFO - 17:25:59: ...  78%|███████▊  | 7/9 [00:00<00:00, 636.95 it/sec, obj=4]
    INFO - 17:25:59: ...  89%|████████▉ | 8/9 [00:00<00:00, 666.48 it/sec, obj=5]
    INFO - 17:25:59: ... 100%|██████████| 9/9 [00:00<00:00, 693.35 it/sec, obj=6]
    INFO - 17:25:59: Optimization result:
    INFO - 17:25:59:    Optimizer info:
    INFO - 17:25:59:       Status: None
    INFO - 17:25:59:       Message: None
    INFO - 17:25:59:       Number of calls to the objective function by the optimizer: 9
    INFO - 17:25:59:    Solution:
    INFO - 17:25:59:       Objective: 1.0
    INFO - 17:25:59:       Design space:
    INFO - 17:25:59:       +------+-------------+-------+-------------+-------+
    INFO - 17:25:59:       | name | lower_bound | value | upper_bound | type  |
    INFO - 17:25:59:       +------+-------------+-------+-------------+-------+
    INFO - 17:25:59:       | x_1  |      0      |   0   |      1      | float |
    INFO - 17:25:59:       | x_2  |      0      |   0   |      1      | float |
    INFO - 17:25:59:       +------+-------------+-------+-------------+-------+
    INFO - 17:25:59: *** End DOEScenario execution (time: 0:00:00.028922) ***

{'eval_jac': False, 'algo': 'fullfact', 'n_samples': 9}

Create the regression model

Then, we build the linear regression model from the database and displays this model.

dataset = scenario.export_to_dataset(opt_naming=False)
model = create_regression_model("RBFRegressor", data=dataset)
model.learn()
print(model)

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/4.2.0/lib/python3.9/site-packages/gemseo/mlearning/transform/scaler/min_max_scaler.py:73: RuntimeWarning: divide by zero encountered in divide
  self.coefficient = where(is_constant, nan_to_num(1 / l_b), 1 / delta)
RBFRegressor(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)

{'y_1': array([6.45029404])}