Test Functions for Metamodeling

The table below listed the available test functions typically used in the comparison of metamodeling approaches.

Name	Input Dimension	Constructor
Ackley	M	Ackley()
Alemazkoor & Meidani (2018) 2D	2	Alemazkoor2D()
Alemazkoor & Meidani (2018) 20D	20	Alemazkoor20D()
Borehole	8	Borehole()
Cheng and Sandu (2010) 2D	2	Cheng2D
Coffee Cup Model	2	CoffeeCup()
Currin et al. (1988) Sine	1	CurrinSine()
Damped Cosine	1	DampedCosine()
Damped Oscillator	7	DampedOscillator()
Dette & Pepelyshev (2010) 8D	3	Dette8D()
Dette & Pepelyshev (2010) Curved	3	DetteCurved()
Dette & Pepelyshev (2010) Exponential	3	DetteExp()
Flood	8	Flood()
Forrester et al. (2008)	1	Forrester2008()
(1st) Franke	2	Franke1()
(2nd) Franke	2	Franke2()
(3rd) Franke	2	Franke3()
(4th) Franke	2	Franke4()
(5th) Franke	2	Franke5()
(6th) Franke	2	Franke6()
Friedman (6D)	6	Friedman6D()
Friedman (10D)	10	Friedman10D()
Genz (Corner Peak)	M	GenzCornerPeak()
Gramacy (2007) Sine	1	GramacySine()
Higdon (2002) Sine	1	HigdonSine()
Holsclaw et al. (2013) Sine	1	HolsclawSine()
Lim et al. (2002) Non-Polynomial	2	LimNonPoly()
Lim et al. (2002) Polynomial	2	LimPoly()
Linkletter et al. (2006) Decreasing Coefficients	10	LinkletterDecCoeffs()
Linkletter et al. (2006) Linear	10	LinkletterLinear()
Linkletter et al. (2006) Sine	10	LinkletterSine()
McLain S1	2	McLainS1()
McLain S2	2	McLainS2()
McLain S3	2	McLainS3()
McLain S4	2	McLainS4()
McLain S5	2	McLainS5()
Oakley & O’Hagan (2002) 1D	1	Oakley1D()
OTL Circuit	6 / 20	OTLCircuit()
Piston Simulation	7 / 20	Piston()
Robot Arm	8	RobotArm()
Rosenbrock	M	Rosenbrock()
Solar Cell Model	5	SolarCell()
Sulfur	9	Sulfur()
Undamped Oscillator	6	UndampedOscillator()
Webster et al. (1996) 2D	2	Webster2D()
Welch et al. (1992)	20	Welch1992()
Wing Weight	10	WingWeight()

In a Python terminal, you can list all the available functions relevant for metamodeling applications using list_functions() and filter the results using the tag parameter (shown below in the HTML format):

import uqtestfuns as uqtf

uqtf.list_functions(tag="metamodeling", tablefmt="html")

No.	Constructor	# Input	# Output	Param.	Description
1	Ackley()	M	1	True	Optimization test function from Ackley (1987)
2	Alemazkoor20D()	20	1	False	High-dimensional low-degree polynomial from Alemazkoor & Meidani (2018)
3	Alemazkoor2D()	2	1	False	Low-dimensional high-degree polynomial from Alemazkoor & Meidani (2018)
4	Borehole()	8	1	False	Borehole function from Harper and Gupta (1983)
5	Cheng2D()	2	1	False	Two-dimensional test function from Cheng and Sandu (2010)
6	CoffeeCup()	2	150	True	Cooling coffee cup model from Tennøe et al. (2018)
7	CurrinSine()	1	1	False	Sine function from Currin et al. (1988)
8	DampedCosine()	1	1	False	One-dimensional damped cosine from Santner et al. (2018)
9	DampedOscillator()	7	1	False	Damped oscillator model from Igusa and Der Kiureghian (1985)
10	Dette8D()	8	1	False	8D function from Dette and Pepelyshev (2010)
11	DetteCurved()	3	1	False	Curved function from Dette and Pepelyshev (2010)
12	DetteExp()	3	1	False	Exponential function from Dette and Pepelyshev (2010)
13	Flood()	8	1	False	Flood model from Iooss and Lemaître (2015)
14	Forrester2008()	1	1	False	One-dimensional function from Forrester et al. (2008)
15	Franke1()	2	1	False	(1st) Franke function from Franke (1979)
16	Franke2()	2	1	False	(2nd) Franke function from Franke (1979)
17	Franke3()	2	1	False	(3rd) Franke function from Franke (1979)
18	Franke4()	2	1	False	(4th) Franke function from Franke (1979)
19	Franke5()	2	1	False	(5th) Franke function from Franke (1979)
20	Franke6()	2	1	False	(6th) Franke function from Franke (1979)
21	Friedman10D()	10	1	False	Ten-dimensional function from Friedman (1991)
22	Friedman6D()	6	1	False	Six-dimensional function from Friedman et al. (1983)
23	GenzCornerPeak()	M	1	True	Corner peak integrand from Genz (1984)
24	GramacySine()	1	1	False	One-dimensional sine function from Gramacy (2007)
25	HigdonSine()	1	1	False	Sine function from Higdon (2002)
26	HolsclawSine()	1	1	False	Sine function from Holsclaw et al. (2013)
27	LimNonPoly()	2	1	False	Two-dimensional non-polynomial function from Lim et al. (2002)
28	LimPoly()	2	1	False	Two-dimensional polynomial function from Lim et al. (2002)
29	LinkletterDecCoeffs()	10	1	False	Linear function with decreasing coefficients (8 active inputs) from Linkletter et al. (2006)
30	LinkletterLinear()	10	1	False	Linear function with 4 active inputs from Linkletter et al. (2006)
31	LinkletterSine()	10	1	False	Sine function with 2 active inputs from Linkletter et al. (2006)
32	McLainS1()	2	1	False	McLain S1 function from McLain (1974)
33	McLainS2()	2	1	False	McLain S2 function from McLain (1974)
34	McLainS3()	2	1	False	McLain S3 function from McLain (1974)
35	McLainS4()	2	1	False	McLain S4 function from McLain (1974)
36	McLainS5()	2	1	False	McLain S5 function from McLain (1974)
37	OTLCircuit()	6	1	False	Output transformerless (OTL) circuit model from Ben-Ari and Steinberg (2007)
38	Oakley1D()	1	1	False	One-dimensional function from Oakley and O'Hagan (2002)
39	Piston()	7	1	False	Piston simulation model from Ben-Ari and Steinberg (2007)
40	RobotArm()	8	1	False	Four-segment robot arm function from An and Owen (2001)
41	Rosenbrock()	M	1	True	Optimization test function from Rosenbrock (1960), also known as the banana function
42	SolarCell()	5	1	True	Single-diode solar-cell model from Constantine et al. (2015)
43	Sulfur()	9	1	False	Sulfur model from Charlson et al. (1992)
44	UndampedOscillator()	6	1	False	Undamped, non-linear, single DOF oscillator
45	Webster2D()	2	1	False	2D polynomial function from Webster et al. (1996).
46	Welch1992()	20	1	False	20-Dimensional function from Welch et al. (1992)
47	WingWeight()	10	1	False	Wing weight model from Forrester et al. (2008)

About metamodeling

In practice, the computational model \(\mathcal{M}\) (see Uncertainty Quantification Framework) is typically complex. Since an uncertainty quantification (UQ) analysis usually require numerous evaluations of \(\mathcal{M}\) (\(\sim 10^2\)—\(10^6\) or more!), the process may become computationally intractable if \(\mathcal{M}\) is expensive to evaluate.

To address this challenge, many UQ analyses employ a metamodel (surrogate model). Constructed on a limited number of model \(\mathcal{M}\) evaluations, such a metamodel should be able to capture the most important aspects of the input/output mapping while being significantly cheaper to evaluate. This metamodel can then replace \(\mathcal{M}\) in the analysis, providing a significant reduction in computational cost without sacrificing the accuracy of the analysis much.

While not a goal of UQ analyse per se, metamodeling is nowadays an indispensable component of the UQ framework [Sud12, SMW17].

References

[Sud12]

Bruno Sudret. Meta-models for structural reliability and uncertainty quantification. In Proceedings of the 5th Asian-Pacific Symposium on Structural Reliability and its Applications. 2012. doi:10.3850/978-981-07-2219-7_p321.

[SMW17]

Bruno Sudret, Stefano Marelli, and Joe Wiart. Surrogate models for uncertainty quantification: an overview. In 2017 11th European Conference on Antennas and Propagation (EUCAP), 793–797. IEEE, March 2017. doi:10.23919/eucap.2017.7928679.