Polynomial Chaos Expansion Construction

This tutorial demonstrates how to create a Polynomial Chaos Expansion (PCE) surrogate model for scalar valued functions. We will use the pytuq.surrogates.pce wrapper class to approximate the function \(\sin^4(x)\).

PyTUQ provides a number of utilities for your workflows, including modules for mapping, test functions, and metric comparisons – these, along with others, can be found in the utils directory. You can also explore the func directory for additional sample functions to use.

This example below outlines how to:

• Define basic parameters for the surrogate model

• Use a sample function of \(\sin^4(x)\) from the utils directory

• Set up a PCE surrogate model

• And evaluate the performance of your model

import numpy as np
import pytuq.utils.funcbank as fcb


from pytuq.surrogates.pce import PCE
from pytuq.utils.maps import scale01ToDom
from pytuq.lreg.anl import anl
from pytuq.lreg.lreg import lsq
from pytuq.utils.mindex import get_mi

N = 14                      # Number of data points to generate
order = 4                   # Polynomial order
true_model = fcb.sin4       # Function to approximate
dim = 3                     # Dimensionality of input data
# dim = 1

np.random.seed(42)

# Domain: defining range of input variable
domain = np.ones((dim, 2))*np.array([-1.,1.])

# Generating x-y data
x = scale01ToDom(np.random.rand(N,dim), domain)
y = true_model(x)

# Testing PCE class:

# (1) Construct polynomial chaos expansion
pce = PCE(dim, order, 'LU')
# pce = PCE(dim, order, 'HG')
# pce = PCE(dim, order, ['LU', 'HG']) # dim = 2
# pce = PCE(dim, order, ['HG', 'LU', 'HG', 'LU']) # dim = 4

pce.set_training_data(x, y)

# (2) Pick method for linear regression object, defaulting to least squares regression
# print(pce.build(regression = 'anl'))
# print(pce.build(regression = 'anl', method = 'vi'))
print(pce.build())
# print(pce.build(regression = 'lsq'))
# print(pce.build(regression = 'bcs'))

# (3) Make predictions for data points and print results:
results = pce.evaluate(x)

# np.random.seed(45) # Generate a single random data point within the domain:
# single_point = scale01ToDom(np.random.rand(1, dim), domain)
# results = pce.evaluate(single_point)

# Generate random input for evaluation:
# x_eval = scale01ToDom(np.random.rand(10,dim), domain)
# results = pce.evaluate(x_eval)

print(results)