Note
Go to the end to download the full example code.
Polynomial chaos expansion (PCE)#
A PCERegressor is a PCE model
based on OpenTURNS.
from __future__ import annotations
from matplotlib import pyplot as plt
from numpy import array
from gemseo import configure_logger
from gemseo import create_discipline
from gemseo import create_parameter_space
from gemseo import sample_disciplines
from gemseo.mlearning import create_regression_model
configure_logger()
<RootLogger root (INFO)>
Problem#
In this example,
we represent the function \(f(x)=(6x-2)^2\sin(12x-4)\) [FSK08]
by the AnalyticDiscipline
discipline = create_discipline(
"AnalyticDiscipline",
name="f",
expressions={"y": "(6*x-2)**2*sin(12*x-4)"},
)
and seek to approximate it over the input space
input_space = create_parameter_space()
input_space.add_random_variable("x", "OTUniformDistribution")
To do this, we create a training dataset with 6 equispaced points:
training_dataset = sample_disciplines(
[discipline], input_space, "y", algo_name="PYDOE_FULLFACT", n_samples=10
)
WARNING - 00:09:41: No coupling in MDA, switching chain_linearize to True.
INFO - 00:09:41: *** Start Sampling execution ***
INFO - 00:09:41: Sampling
INFO - 00:09:41: Disciplines: f
INFO - 00:09:41: MDO formulation: MDF
INFO - 00:09:41: Running the algorithm PYDOE_FULLFACT:
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INFO - 00:09:41: *** End Sampling execution (time: 0:00:00.005014) ***
Basics#
Training#
Then, we train an PCE regression model from these samples:
model = create_regression_model("PCERegressor", training_dataset)
model.learn()
WARNING - 00:09:41: Remove input data transformation because PCERegressor does not support transformers.
Prediction#
Once it is built, we can predict the output value of \(f\) at a new input point:
input_value = {"x": array([0.65])}
output_value = model.predict(input_value)
output_value
{'y': array([-0.81106394])}
as well as its Jacobian value:
jacobian_value = model.predict_jacobian(input_value)
jacobian_value
{'y': {'x': array([[18.2279622]])}}
Plotting#
Of course, you can see that the quadratic model is no good at all here:
test_dataset = sample_disciplines(
[discipline], input_space, "y", algo_name="PYDOE_FULLFACT", n_samples=100
)
input_data = test_dataset.get_view(variable_names=model.input_names).to_numpy()
reference_output_data = test_dataset.get_view(variable_names="y").to_numpy().ravel()
predicted_output_data = model.predict(input_data).ravel()
plt.plot(input_data.ravel(), reference_output_data, label="Reference")
plt.plot(input_data.ravel(), predicted_output_data, label="Regression - Basics")
plt.grid()
plt.legend()
plt.show()
WARNING - 00:09:41: No coupling in MDA, switching chain_linearize to True.
INFO - 00:09:41: *** Start Sampling execution ***
INFO - 00:09:41: Sampling
INFO - 00:09:41: Disciplines: f
INFO - 00:09:41: MDO formulation: MDF
INFO - 00:09:41: Running the algorithm PYDOE_FULLFACT:
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INFO - 00:09:41: *** End Sampling execution (time: 0:00:00.025480) ***
Settings#
The PCERegressor has many options
defined in the PCERegressor_Settings Pydantic model.
Degree#
model = create_regression_model("PCERegressor", training_dataset, degree=3)
model.learn()
WARNING - 00:09:41: Remove input data transformation because PCERegressor does not support transformers.
and see that this model seems to be better:
predicted_output_data_ = model.predict(input_data).ravel()
plt.plot(input_data.ravel(), reference_output_data, label="Reference")
plt.plot(input_data.ravel(), predicted_output_data, label="Regression - Basics")
plt.plot(input_data.ravel(), predicted_output_data_, label="Regression - Degree(3)")
plt.grid()
plt.legend()
plt.show()
Total running time of the script: (0 minutes 0.192 seconds)