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 a MLRegressionAlgo

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.api import create_surrogate
from numpy import array

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 - 16:57:54:
    INFO - 16:57:54: *** Start DOEScenario execution ***
    INFO - 16:57:54: DOEScenario
    INFO - 16:57:54:    Disciplines: func
    INFO - 16:57:54:    MDO formulation: DisciplinaryOpt
    INFO - 16:57:54: Optimization problem:
    INFO - 16:57:54:    minimize y_1(x_1, x_2)
    INFO - 16:57:54:    with respect to x_1, x_2
    INFO - 16:57:54:    over the design space:
    INFO - 16:57:54:    +------+-------------+-------+-------------+-------+
    INFO - 16:57:54:    | name | lower_bound | value | upper_bound | type  |
    INFO - 16:57:54:    +------+-------------+-------+-------------+-------+
    INFO - 16:57:54:    | x_1  |      0      |  None |      1      | float |
    INFO - 16:57:54:    | x_2  |      0      |  None |      1      | float |
    INFO - 16:57:54:    +------+-------------+-------+-------------+-------+
    INFO - 16:57:54: Solving optimization problem with algorithm fullfact:
    INFO - 16:57:54: ...   0%|          | 0/9 [00:00<?, ?it]
    INFO - 16:57:54: ...  11%|█         | 1/9 [00:00<00:00, 351.08 it/sec, obj=1]
    INFO - 16:57:54: ...  22%|██▏       | 2/9 [00:00<00:00, 582.58 it/sec, obj=2]
    INFO - 16:57:54: ...  33%|███▎      | 3/9 [00:00<00:00, 760.71 it/sec, obj=3]
    INFO - 16:57:54: ...  44%|████▍     | 4/9 [00:00<00:00, 898.57 it/sec, obj=2.5]
    INFO - 16:57:54: ...  56%|█████▌    | 5/9 [00:00<00:00, 995.80 it/sec, obj=3.5]
    INFO - 16:57:54: ...  67%|██████▋   | 6/9 [00:00<00:00, 1068.52 it/sec, obj=4.5]
    INFO - 16:57:54: ...  78%|███████▊  | 7/9 [00:00<00:00, 1123.23 it/sec, obj=4]
    INFO - 16:57:54: ...  89%|████████▉ | 8/9 [00:00<00:00, 1186.89 it/sec, obj=5]
    INFO - 16:57:54: ... 100%|██████████| 9/9 [00:00<00:00, 1242.14 it/sec, obj=6]
    INFO - 16:57:54: Optimization result:
    INFO - 16:57:54:    Optimizer info:
    INFO - 16:57:54:       Status: None
    INFO - 16:57:54:       Message: None
    INFO - 16:57:54:       Number of calls to the objective function by the optimizer: 9
    INFO - 16:57:54:    Solution:
    INFO - 16:57:54:       Objective: 1.0
    INFO - 16:57:54:       Design space:
    INFO - 16:57:54:       +------+-------------+-------+-------------+-------+
    INFO - 16:57:54:       | name | lower_bound | value | upper_bound | type  |
    INFO - 16:57:54:       +------+-------------+-------+-------------+-------+
    INFO - 16:57:54:       | x_1  |      0      |   0   |      1      | float |
    INFO - 16:57:54:       | x_2  |      0      |   0   |      1      | float |
    INFO - 16:57:54:       +------+-------------+-------+-------------+-------+
    INFO - 16:57:54: *** End DOEScenario execution (time: 0:00:00.015995) ***

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

Create the surrogate discipline

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

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

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/4.3.0.post0/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)
    INFO - 16:57:55: Build the surrogate discipline: GPR_DOEScenario
    INFO - 16:57:55:    Dataset name: DOEScenario
    INFO - 16:57:55:    Dataset size: 9
    INFO - 16:57:55:    Surrogate model: GaussianProcessRegressor
    INFO - 16:57:55: Use the surrogate discipline: GPR_DOEScenario
    INFO - 16:57:55:    Inputs: x_1, x_2
    INFO - 16:57:55:    Outputs: y_1
    INFO - 16:57:55:    Jacobian: use finite differences

Predict output

Once it is built, we can use it for prediction, either with default inputs or with user-defined ones.

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

{'x_1': array([0.5]), 'x_2': array([0.5]), 'y_1': array([3.49999999])}
{'x_1': array([1.]), 'x_2': array([2.]), 'y_1': array([8.50166061])}