Exponential function (3 random inputs, scalar output)

In this example, PCE is used to generate a surrogate model for a given set of 3D data.

Dette & Pepelyshev exponential function

\[f(x) = 100(\exp{(-2/x_1^{1.75})} + \exp{(-2/x_2^{1.5})} + \exp{(-2/x_3^{1.25})})\]

Description: Dimensions: 3

Input Domain: This function is evaluated on the hypercube \(x_i \in [0,1]\) for all \(i = 1,2,3\).

Reference: Dette, H., & Pepelyshev, A. (2010). Generalized Latin hypercube design for computer experiments. Technometrics, 52(4).

Import necessary libraries.

import math
import numpy as np
import matplotlib.pyplot as plt
from UQpy.distributions import Uniform, JointIndependent
from UQpy.surrogates import *

Define the function.

def function(x):
    return 100*(np.exp(-2/(x[:,0]**1.75)) + np.exp(-2/(x[:,1]**1.5)) + np.exp(-2/(x[:,2]**1.25)))

Define the input probability distributions.

# input distributions
dist = Uniform(loc=0, scale=1)
marg = [dist]*3
joint = JointIndependent(marginals=marg)

Compute reference mean and variance values using Monte Carlo sampling.

# reference moments via Monte Carlo Sampling
n_samples_mc = 1000000
xx = joint.rvs(n_samples_mc)
yy = function(xx)
mean_ref = yy.mean()
var_ref = yy.var()

Create validation data sets, to be used later to estimate the accuracy of the PCE.

# validation data sets
n_samples_val = 1000
xx_val = joint.rvs(n_samples_val)
yy_val = function(xx_val)

Assess the PCE in terms of approximation and moment estimation accuracy, for increasing maximum polynomial degree.

# construct PCE surrogate models
l2_err = []
mean_err = []
var_err = []
for max_degree in range(1, 10):
    print(' ')

    # PCE basis
    print('Total degree: ', max_degree)
    polynomial_basis = TotalDegreeBasis(joint, max_degree)
    print('Size of basis:', polynomial_basis.polynomials_number)

    # generate training data
    sampling_coeff = 5
    print('Sampling coefficient: ', sampling_coeff)
    np.random.seed(42)
    n_samples = math.ceil(sampling_coeff * polynomial_basis.polynomials_number)
    print('Training data: ', n_samples)
    xx_train = joint.rvs(n_samples)
    yy_train = function(xx_train)

    # fit model
    least_squares = LeastSquareRegression()
    pce_metamodel = PolynomialChaosExpansion(polynomial_basis=polynomial_basis, regression_method=least_squares)
    pce_metamodel.fit(xx_train, yy_train)

    # validation errors
    yy_val_pce = pce_metamodel.predict(xx_val)
    errors = np.abs(yy_val - yy_val_pce.flatten())
    error_l2 = np.linalg.norm(errors)
    l2_err.append(error_l2)
    print('Validation error in the l2 norm:', error_l2)

    # moment errors
    pce_moments = pce_metamodel.get_moments()
    mean_pce = pce_moments[0]
    var_pce = pce_moments[1]
    mean_err.append(np.abs((mean_pce - mean_ref) / mean_ref))
    var_err.append(np.abs((var_pce - var_ref) / var_ref))
    print('Relative error in the mean:', mean_err[-1])
    print('Relative error in the variance:', var_err[-1])

Total degree:  1
Size of basis: 4
Sampling coefficient:  5
Training data:  20
Validation error in the l2 norm: 129.20598047202918
Relative error in the mean: 0.06718582089180117
Relative error in the variance: 0.19464294065482993

Total degree:  2
Size of basis: 10
Sampling coefficient:  5
Training data:  50
Validation error in the l2 norm: 32.491868013742526
Relative error in the mean: 0.005657152666309865
Relative error in the variance: 0.021441707594204832

Total degree:  3
Size of basis: 20
Sampling coefficient:  5
Training data:  100
Validation error in the l2 norm: 11.203232601483554
Relative error in the mean: 0.0037013939722399386
Relative error in the variance: 0.028353176125452285

Total degree:  4
Size of basis: 35
Sampling coefficient:  5
Training data:  175
Validation error in the l2 norm: 6.491941276358539
Relative error in the mean: 0.001343264325296654
Relative error in the variance: 0.0028847488197525036

Total degree:  5
Size of basis: 56
Sampling coefficient:  5
Training data:  280
Validation error in the l2 norm: 2.109018991714873
Relative error in the mean: 0.0020302227928515534
Relative error in the variance: 0.0030831698249971394

Total degree:  6
Size of basis: 84
Sampling coefficient:  5
Training data:  420
Validation error in the l2 norm: 1.51701246194853
Relative error in the mean: 0.0014108145347999925
Relative error in the variance: 0.004548486290468092

Total degree:  7
Size of basis: 120
Sampling coefficient:  5
Training data:  600
Validation error in the l2 norm: 0.6558976196693772
Relative error in the mean: 0.0011112582388267986
Relative error in the variance: 0.0015059072771149253

Total degree:  8
Size of basis: 165
Sampling coefficient:  5
Training data:  825
Validation error in the l2 norm: 0.3622570432474069
Relative error in the mean: 0.0012297780921141103
Relative error in the variance: 0.0021545930796069593

Total degree:  9
Size of basis: 220
Sampling coefficient:  5
Training data:  1100
Validation error in the l2 norm: 0.14772577969491651
Relative error in the mean: 0.0011826721179544438
Relative error in the variance: 0.0022894928914782933