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 create_discipline
from gemseo import create_parameter_space
from gemseo import sample_disciplines
from gemseo.mlearning import create_regression_model

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
)

INFO - 16:19:13: *** Start Sampling execution ***
INFO - 16:19:13: Sampling
INFO - 16:19:13:    Disciplines: f
INFO - 16:19:13:    MDO formulation: MDF
INFO - 16:19:13: Running the algorithm PYDOE_FULLFACT:
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INFO - 16:19:13: *** End Sampling execution ***

Basics

Training

Then, we train an PCE regression model from these samples:

model = create_regression_model("PCERegressor", training_dataset)
model.learn()

WARNING - 16:19:13: 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()

INFO - 16:19:13: *** Start Sampling execution ***
INFO - 16:19:13: Sampling
INFO - 16:19:13:    Disciplines: f
INFO - 16:19:13:    MDO formulation: MDF
INFO - 16:19:13: Running the algorithm PYDOE_FULLFACT:
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INFO - 16:19:13: *** End Sampling execution ***

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 - 16:19:13: 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()