Create a surrogate discipline

We want to build a 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 division, unicode_literals

Import

from numpy import array

from gemseo.api import (
    configure_logger,
    create_design_space,
    create_discipline,
    create_scenario,
    create_surrogate,
)

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()
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.

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 - 09:24:11:
    INFO - 09:24:11: *** Start DOE Scenario execution ***
    INFO - 09:24:11: DOEScenario
    INFO - 09:24:11:    Disciplines: func
    INFO - 09:24:11:    MDOFormulation: DisciplinaryOpt
    INFO - 09:24:11:    Algorithm: fullfact
    INFO - 09:24:11: Optimization problem:
    INFO - 09:24:11:    Minimize: y_1(x_1, x_2)
    INFO - 09:24:11:    With respect to: x_1, x_2
    INFO - 09:24:11: Full factorial design required. Number of samples along each direction for a design vector of size 2 with 9 samples: 3
    INFO - 09:24:11: Final number of samples for DOE = 9 vs 9 requested
    INFO - 09:24:11: DOE sampling:   0%|          | 0/9 [00:00<?, ?it]
    INFO - 09:24:11: DOE sampling: 100%|██████████| 9/9 [00:00<00:00, 408.28 it/sec, obj=6]
    INFO - 09:24:11: Optimization result:
    INFO - 09:24:11: Objective value = 1.0
    INFO - 09:24:11: The result is feasible.
    INFO - 09:24:11: Status: None
    INFO - 09:24:11: Optimizer message: None
    INFO - 09:24:11: Number of calls to the objective function by the optimizer: 9
    INFO - 09:24:11: Design Space:
    INFO - 09:24:11: +------+-------------+-------+-------------+-------+
    INFO - 09:24:11: | name | lower_bound | value | upper_bound | type  |
    INFO - 09:24:11: +------+-------------+-------+-------------+-------+
    INFO - 09:24:11: | x_1  |      0      |   0   |      1      | float |
    INFO - 09:24:11: | x_2  |      0      |   0   |      1      | float |
    INFO - 09:24:11: +------+-------------+-------+-------------+-------+
    INFO - 09:24:11: *** DOE Scenario run terminated ***

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

Create the surrogate discipline

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

dataset = discipline.cache.export_to_dataset()
model = create_surrogate("GaussianProcessRegression", data=dataset)

Out:

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.1.0/lib/python3.8/site-packages/sklearn/gaussian_process/_gpr.py:506: ConvergenceWarning: lbfgs failed to converge (status=2):
ABNORMAL_TERMINATION_IN_LNSRCH.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.org/stable/modules/preprocessing.html
  _check_optimize_result("lbfgs", opt_res)
    INFO - 09:24:11: Build the surrogate discipline: GPR_func
    INFO - 09:24:11:    Dataset name: func
    INFO - 09:24:11:    Dataset size: 9
    INFO - 09:24:11:    Surrogate model: GaussianProcessRegression
    INFO - 09:24:11: Use the surrogate discipline: GPR_func
    INFO - 09:24:11:    Inputs: x_1, x_2
    INFO - 09:24:11:    Outputs: y_1, y_2
    INFO - 09:24:11:    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)

Out:

{'x_1': array([0.5]), 'x_2': array([0.5]), 'y_1': array([3.49999999]), 'y_2': array([-3.50000001])}
{'x_1': array([1.]), 'x_2': array([2.]), 'y_1': array([8.50166078]), 'y_2': array([-8.56035203])}