Create a surrogate discipline

We want to build an MDODiscipline based on a regression model approximating the following 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 \([0,1]\times[0,1]\). For that, we use a SurrogateDiscipline relying on an MLRegressionAlgo

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 import create_surrogate

Import

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 - 18:54:58:
    INFO - 18:54:58: *** Start DOEScenario execution ***
    INFO - 18:54:58: DOEScenario
    INFO - 18:54:58:    Disciplines: func
    INFO - 18:54:58:    MDO formulation: DisciplinaryOpt
    INFO - 18:54:58: Optimization problem:
    INFO - 18:54:58:    minimize y_1(x_1, x_2)
    INFO - 18:54:58:    with respect to x_1, x_2
    INFO - 18:54:58:    over the design space:
    INFO - 18:54:58:    +------+-------------+-------+-------------+-------+
    INFO - 18:54:58:    | name | lower_bound | value | upper_bound | type  |
    INFO - 18:54:58:    +------+-------------+-------+-------------+-------+
    INFO - 18:54:58:    | x_1  |      0      |  None |      1      | float |
    INFO - 18:54:58:    | x_2  |      0      |  None |      1      | float |
    INFO - 18:54:58:    +------+-------------+-------+-------------+-------+
    INFO - 18:54:58: Solving optimization problem with algorithm fullfact:
    INFO - 18:54:58: ...  11%|█         | 1/9 [00:00<00:00, 321.45 it/sec, obj=1]
    INFO - 18:54:58: ...  22%|██▏       | 2/9 [00:00<00:00, 533.36 it/sec, obj=2]
    INFO - 18:54:58: ...  33%|███▎      | 3/9 [00:00<00:00, 695.15 it/sec, obj=3]
    INFO - 18:54:58: ...  44%|████▍     | 4/9 [00:00<00:00, 814.94 it/sec, obj=2.5]
    INFO - 18:54:58: ...  56%|█████▌    | 5/9 [00:00<00:00, 915.59 it/sec, obj=3.5]
    INFO - 18:54:58: ...  67%|██████▋   | 6/9 [00:00<00:00, 998.49 it/sec, obj=4.5]
    INFO - 18:54:58: ...  78%|███████▊  | 7/9 [00:00<00:00, 1067.99 it/sec, obj=4]
    INFO - 18:54:58: ...  89%|████████▉ | 8/9 [00:00<00:00, 1123.99 it/sec, obj=5]
    INFO - 18:54:58: ... 100%|██████████| 9/9 [00:00<00:00, 1173.56 it/sec, obj=6]
    INFO - 18:54:58: Optimization result:
    INFO - 18:54:58:    Optimizer info:
    INFO - 18:54:58:       Status: None
    INFO - 18:54:58:       Message: None
    INFO - 18:54:58:       Number of calls to the objective function by the optimizer: 9
    INFO - 18:54:58:    Solution:
    INFO - 18:54:58:       Objective: 1.0
    INFO - 18:54:58:       Design space:
    INFO - 18:54:58:       +------+-------------+-------+-------------+-------+
    INFO - 18:54:58:       | name | lower_bound | value | upper_bound | type  |
    INFO - 18:54:58:       +------+-------------+-------+-------------+-------+
    INFO - 18:54:58:       | x_1  |      0      |   0   |      1      | float |
    INFO - 18:54:58:       | x_2  |      0      |   0   |      1      | float |
    INFO - 18:54:58:       +------+-------------+-------+-------------+-------+
    INFO - 18:54:58: *** End DOEScenario execution (time: 0:00:00.018441) ***

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

Create the surrogate discipline

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

dataset = scenario.to_dataset(opt_naming=False)
model = create_surrogate("GaussianProcessRegressor", data=dataset)

INFO - 18:54:58: Build the surrogate discipline: GPR_DOEScenario
INFO - 18:54:58:    Dataset size: 9
INFO - 18:54:58:    Surrogate model: GaussianProcessRegressor
INFO - 18:54:58: Use the surrogate discipline: GPR_DOEScenario
INFO - 18:54:58:    Inputs: x_1, x_2
INFO - 18:54:58:    Outputs: y_1
INFO - 18:54:58:    Jacobian: use finite differences

Predict output

Once it is built, we can use it for prediction, either with default inputs

model.execute()

{'x_1': array([0.5]), 'x_2': array([0.5]), 'y_1': array([3.49999999])}

or with user-defined ones.

model.execute({"x_1": array([1.0]), "x_2": array([2.0])})

{'x_1': array([1.]), 'x_2': array([2.]), 'y_1': array([8.50166033])}