Note
Go to the end to download the full example code
PCE 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 import configure_logger
from gemseo import create_design_space
from gemseo import create_discipline
from gemseo import create_parameter_space
from gemseo import create_scenario
from gemseo.mlearning import create_regression_model
from gemseo.mlearning import import_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 - 07:05:46:
INFO - 07:05:46: *** Start DOEScenario execution ***
INFO - 07:05:46: DOEScenario
INFO - 07:05:46: Disciplines: func
INFO - 07:05:46: MDO formulation: DisciplinaryOpt
INFO - 07:05:46: Optimization problem:
INFO - 07:05:46: minimize y_1(x_1, x_2)
INFO - 07:05:46: with respect to x_1, x_2
INFO - 07:05:46: over the design space:
INFO - 07:05:46: +------+-------------+-------+-------------+-------+
INFO - 07:05:46: | name | lower_bound | value | upper_bound | type |
INFO - 07:05:46: +------+-------------+-------+-------------+-------+
INFO - 07:05:46: | x_1 | 0 | None | 1 | float |
INFO - 07:05:46: | x_2 | 0 | None | 1 | float |
INFO - 07:05:46: +------+-------------+-------+-------------+-------+
INFO - 07:05:46: Solving optimization problem with algorithm fullfact:
INFO - 07:05:46: ... 0%| | 0/9 [00:00<?, ?it]
INFO - 07:05:46: ... 11%|█ | 1/9 [00:00<00:00, 332.17 it/sec, obj=1]
INFO - 07:05:46: ... 22%|██▏ | 2/9 [00:00<00:00, 546.17 it/sec, obj=2]
INFO - 07:05:46: ... 33%|███▎ | 3/9 [00:00<00:00, 704.73 it/sec, obj=3]
INFO - 07:05:46: ... 44%|████▍ | 4/9 [00:00<00:00, 813.01 it/sec, obj=2.5]
INFO - 07:05:46: ... 56%|█████▌ | 5/9 [00:00<00:00, 901.38 it/sec, obj=3.5]
INFO - 07:05:46: ... 67%|██████▋ | 6/9 [00:00<00:00, 977.09 it/sec, obj=4.5]
INFO - 07:05:46: ... 78%|███████▊ | 7/9 [00:00<00:00, 1036.54 it/sec, obj=4]
INFO - 07:05:46: ... 89%|████████▉ | 8/9 [00:00<00:00, 1090.28 it/sec, obj=5]
INFO - 07:05:46: ... 100%|██████████| 9/9 [00:00<00:00, 1137.32 it/sec, obj=6]
INFO - 07:05:46: Optimization result:
INFO - 07:05:46: Optimizer info:
INFO - 07:05:46: Status: None
INFO - 07:05:46: Message: None
INFO - 07:05:46: Number of calls to the objective function by the optimizer: 9
INFO - 07:05:46: Solution:
INFO - 07:05:46: Objective: 1.0
INFO - 07:05:46: Design space:
INFO - 07:05:46: +------+-------------+-------+-------------+-------+
INFO - 07:05:46: | name | lower_bound | value | upper_bound | type |
INFO - 07:05:46: +------+-------------+-------+-------------+-------+
INFO - 07:05:46: | x_1 | 0 | 0 | 1 | float |
INFO - 07:05:46: | x_2 | 0 | 0 | 1 | float |
INFO - 07:05:46: +------+-------------+-------+-------------+-------+
INFO - 07:05:46: *** End DOEScenario execution (time: 0:00:00.018521) ***
{'eval_jac': False, 'n_samples': 9, 'algo': 'fullfact'}
Create the regression model¶
Then, we build the linear regression model from the database and displays this model.
prob_space = create_parameter_space()
prob_space.add_random_variable("x_1", "OTUniformDistribution")
prob_space.add_random_variable("x_2", "OTUniformDistribution")
dataset = scenario.to_dataset(opt_naming=False)
model = create_regression_model(
"PCERegressor", data=dataset, probability_space=prob_space, transformer=None
)
model.learn()
print(model)
PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=+----------------------------------------------------------------------------------+
| Parameter space |
+------+-------------+-------+-------------+-------+-------------------------------+
| name | lower_bound | value | upper_bound | type | Initial distribution |
+------+-------------+-------+-------------+-------+-------------------------------+
| x_1 | 0 | 0.5 | 1 | float | Uniform(lower=0.0, upper=1.0) |
| x_2 | 0 | 0.5 | 1 | float | Uniform(lower=0.0, upper=1.0) |
+------+-------------+-------+-------------+-------+-------------------------------+, use_cleaning=False, use_lars=False)
based on the OpenTURNS 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([9.])}
Save the regression model¶
Lastly, we save the model.
directory = model.to_pickle()
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/stable/lib/python3.9/site-packages/gemseo/mlearning/core/ml_algo.py:327: UserWarning: Pandas doesn't allow columns to be created via a new attribute name - see https://pandas.pydata.org/pandas-docs/stable/indexing.html#attribute-access
self.learning_set.data = {}
Load the regression model¶
In an other study, we could load this model.
loaded_model = import_regression_model(directory)
print(loaded_model)
PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=+----------------------------------------------------------------------------------+
| Parameter space |
+------+-------------+-------+-------------+-------+-------------------------------+
| name | lower_bound | value | upper_bound | type | Initial distribution |
+------+-------------+-------+-------------+-------+-------------------------------+
| x_1 | 0 | 0.5 | 1 | float | Uniform(lower=0.0, upper=1.0) |
| x_2 | 0 | 0.5 | 1 | float | Uniform(lower=0.0, upper=1.0) |
+------+-------------+-------+-------------+-------+-------------------------------+, use_cleaning=False, use_lars=False)
based on the OpenTURNS library
built from 9 learning samples
Use the loaded regression model¶
And use it!
print(loaded_model.predict(input_value))
{'y_1': array([9.])}
Total running time of the script: ( 0 minutes 0.179 seconds)