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Datasets

Data sets, used for model training and testing, are categorized into two main groups:

Synthetic Data

For synthetic problems, the analytical form of underlying models is known and used to generate data points. This category includes physics equations which are constrained by physical units.
Examples:
mathematical equation: \(f(x) = 2x^2 + \cos(x)\)
physical equation: \(f(x) = Gm_1m_2/r^2\) with \([m_1]=[m_2]=\) Kilograms, \([r]=\) meter and \(G\) is gravitational constant.

Origin
Underlying model
Benchmark name
Reference
problems
year
Physics Ordinary differential equations
General physics equations (classical mechanics, electromagnetism,
quantum mechanics, gravity, nuclear physics, etc. )		 Strogatz
AIFeynman		 Strogatz repositery
Feynman Database		 10
120		 2011
2019



Mathematics


monomials, polynomials,
trigonometric, exponential, etc.		 Koza
Keijer
Vladislavleva
Nguyen
Korns
R
Jin
Livermore		 Koza
Keijer M.
Vladislavleva E.J. et al.
Uy N.Q. et al.
Korns M.F.
Krawiec K., Pawlak T.
Jin Y. et al.
Petersen B. K. et al.		 3
15
8
12
15
3
6
22		 1994
2003
2009
2011
2011
2013
2019
2021
Real Data

For real-world problems, the underlying model is fully unknown. This category includes any type of data such as climate, economics, medical, etc.
Datasets for this category can be found in The Penn Machine Learning benchmarks (PMLB) directory.

Research on SR datasets
Examples of mathematical equations
Dataset
Expression
Variables
Training data range
Koza-1 \(x^4 + x^3 + x^2 + x\) 1 U[-1, 1, 20]
Koza-2 \(x^5 - 2x^3 + x\) 1 U[-1, 1, 20]
Koza-3 \(x^6 - 2x^4 + x^2\) 1 U[-1, 1, 20]
Keijzer-1 \(0.3 x \sin(2\pi x)\) 1 E[-1, 1, 0.1]
Keijzer-2 \(0.3 x \sin(2\pi x)\) 1 E[-2, 2, 0.1]
Keijzer-3 \(0.3 x \sin(2\pi x)\) 1 E[-3, 3, 0.1]
Keijzer-4 \(x^3e^{-x} \cos(x)\sin(x)(\sin^2(x)\cos(x)-1)\) 1 E[0, 10, 0.05]
Keijzer-5 \(30xz/(x-10)y^2\) 3 \(x,z:\) U[-1,1,1000]
\(y:\) U[1,2,1000]
Keijzer-6 \(\sum_1^{x}i\) 1 E[1, 50, 1]
Keijzer-7 \(\log x\) 1 E[1, 100, 1]
Keijzer-8 \(\sqrt{x}\) 1 E[0, 100, 1]
Keijzer-9 \(\mathrm{arcsinh}(x)=\log(x +\sqrt{x^2 + 1})\) 1 E[0, 100, 1]
Keijzer-10 \(x^y\) 2 U[0, 1, 100]
Keijzer-11 \(xy + \sin((x-1)(y-1))\) 2 U[-3, 3, 20]
Keijzer-12 \(x^4-x^3 +y^2/2 - y\) 2 U[-3, 3, 20]
Keijzer-13 \(6\sin(x)\cos(y)\) 2 U[-3, 3, 20]
Keijzer-14 \(8/(2+x^2+y^2)\) 2 U[-3, 3, 20]
Keijzer-15 \(x^3/5 +y^3/2-y-x\) 2 U[-3, 3, 20]
Vladislavleva-1 \(\frac{e^{-(x-1)^2}}{1.2+(y-2.5)^2}\) 1 U[0.3, 4, 100]
Vladislavleva-2 \(e^{-x}x^3(\cos x\sin x)(\cos x \sin^2 x-1)\) 2 E[0.5, 10, 0.1]
Vladislavleva-3 \(e^{-x}x^3(\cos x\sin x)(\cos x\sin^2 x-1)(y-5)\) 2 \(x:\)E[0.05,10,0.1]
\(y:\)E[0.05,10.05,2]
Vladislavleva-4 \(\frac{10}{5+\sum_{i=1}^{5}(x_i-3)^2}\) 5 U[0.05, 6.05, 1024]
Vladislavleva-5 \(30(x-1)\frac{(z-1)}{y^2(x-10)}\) 3 \(x:\) U[0.05, 2, 300]
\(y:\) U[1, 2, 300]
\(z:\) U[0.05, 2, 300]
Vladislavleva-6 \(6\sin(x)\cos(y)\) 2 U[0.1, 5.9, 30]
Vladislavleva-7 \((x-3)(y-3) + 2\sin((x-4)(y-4))\) 2 U[0.05, 6.05, 300]
Vladislavleva-8 \(\frac{(x-3)^4+(y-3)^3-(y-3)}{(y-2)^4+10}\) 2 U[0.05, 6.05, 50]
Nguyen-1 \(x^3+ x^2 + x\) 1 U(-1,1,20)
Nguyen-2 \(x^4 + x^3+ x^2 + x\) 1 U(-1,1,20)
Nguyen-3 \(x^5 + x^4 + x^3+ x^2 + x\) 1 U(-1,1,20)
Nguyen-4 \(x^6 + x^5 + x^4 + x^3+ x^2 + x\) 1 U(-1,1,20)
Nguyen-5 \(\sin(x^2)\cos(x) -1\) 1 U(-1,1,20)
Nguyen-6 \(\sin(x) + \sin(x+x^2)\) 1 U(-1,1,20)
Nguyen-7 \(\log(x+1) + \log(x^2+1)\) 1 U(0,2,20)
Nguyen-8 \(\sqrt{x}\) 1 U(0,4,20)
Nguyen-9 \(\sin(x) + \sin(y^2)\) 2 U(-1,1,100)
Nguyen-10 \(2\sin(x)\cos(y)\) 2 U(-1,1,100)
Nguyen-11 \(x^{y}\) 2
Nguyen-12 \(x^4 - x^3 + \frac{1}{2}y^2 - y\) 2
Korns-1 \(1.57 + (24.3 v)\) 1 U[-50, 50, 10000]
Korns-2 \(0.23 + 14.2\frac{v+y}{3\omega}\) 3 U[-50, 50, 10000]
Korns-3 \(-5.41 + 4.9\frac{v-x+y/w}{3\omega}\) 4 U[-50, 50, 10000]
Korns-4 \(-2.3 + 0.13\sin(z)\) 1 U[-50, 50, 10000]
Korns-5 \(3 + 2.13 \ln(\omega)\) 1 U[-50, 50, 10000]
Korns-6 \(1.3 + 0.13 \sqrt{x}\) 1 U[-50, 50, 10000]
Korns-7 \(213.80940889(1- e^{-0.54723748542 x})\) 1 U[-50, 50, 10000]
Korns-8 \(6.87 + 11 \sqrt{7.23~x~v~\omega}\) 3 U[-50, 50, 10000]
Korns-9 \(\frac{\sqrt{x}}{\ln(y)}\frac{e^z}{v^2}\) 4 U[-50, 50, 10000]
Korns-10 \(0.81 + 24.3\frac{2 y+3 z^2}{4v^3+5\omega^4}\) 4 U[-50, 50, 10000]
Korns-11 \(6.87 + 11\cos(7.23 x^3)\) 1 U[-50, 50, 10000]
Korns-12 \(2-2.1\cos(9.8 x)\sin(1.3\omega)\) 2 U[-50, 50, 10000]
Korns-13 \(32-3\frac{\tan(x)}{\tan(y)}\frac{\tan(z)}{\tan(v)}\) 4 U[-50, 50, 10000]
Korns-14 \(22-4.2(\cos(x)-\tan(y))\frac{\tanh(z)}{\sin(v)}\) 4 U[-50, 50, 10000]
Korns-15 \(12-6\frac{\tan(x)}{e^y}(\ln(z)-\tan(v))\) 4 U[-50, 50, 10000]
R1 \((x+1)^3/(x^2-x+1)\) 1 E[-1,1,20]
R2 \((x^5-3x^3+1)/(x^2+1)\) 1 E[-1,1,20]
R3 \((x^6+x^5)/(x^4+x^3+x^2+x+1)\) 1 E[-1,1,20]
Jin-1 \(2.5x^4 -1.3x^3 +0.5y^2 -1.7y\) 2 U(-3,3,100)
Jin-2 $ 8.0x^2 + 8.0y^3 -15.0$ 2 U(-3,3,100)
Jin-3 $ 0.2x^3 +1.5y^3 -1.2y -0.5x$ 2 U(-3,3,100)
Jin-4 $ 1.5\exp(x) + 5.0\cos(y)$ 2 U(-3,3,100)
Jin-5 $ 6.0\sin(x)\cos(y)$ 2 U(-3,3,100)
Jin-6 $ 1.35xy + 5.5\sin((x-1.0)(y-1.0)$ 2 U(-3,3,100)
Livermore-1 \(1/3 + x + \sin(x^2)\) 1 U[-10,10,1000]
Livermore-2 \(\sin(x^2)\cos(x) - 2\) 1 U[-1,1,20]
Livermore-3 \(\sin(x^3)\cos(x^2) -1\) 1 U[-1,1,20]
Livermore-4 \(\log(x+1) + \log(x^2+1)+\log(x)\) 1 U[0,2,20]
Livermore-5 \(x^4 - x^3 + x^2 -y\) 2 U[0,1,20]
Livermore-6 \(4x^4 + 3x^3 + 2x^2 + x\) 1 U[-1,1,20]
Livermore-7 \(\sinh(x)\) 1 U[-1,1,20]
Livermore-8 \(\cosh(x)\) 1 U[-1,1,20]
Livermore-9 \(x^9 +x^8+x^7+x^6+x^5+x^4+x^3+x^2+x\) 1 U[-1,1,20]
Livermore-10 \(6\sin(x)\cos(y)\) 2 U[0,1,20]
Livermore-11 \(x^2y^2/(x+y)\) 2 U[-1,1,50]
Livermore-12 \(x^5/y^3\) 2 U[-1,1,50]
Livermore-13 \(x^{1/3}\) 1 U[0,4,20]
Livermore-14 \(x^3+x^2+x+\sin(x)+\sin(x^2)\) 1 U[-1,1,20]
Livermore-15 \(x^{1/5}\) 1 U[0,4,20]
Livermore-16 \(x^{2/5}\) 1 U[0,4,20]
Livermore-17 \(4\sin(x)\cos(y)\) 2 U[0,1,20]
Livermore-18 \(\sin(x^2)\cos(x) - 5\) 1 U[-1,1,20]
Livermore-19 \(x^5+x^4+x^2+x\) 1 U[-1,1,20]
Livermore-20 \(\exp(-x^2)\) 1 U[-1,1,20]
Livermore-21 \(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x\) 1 U[-1,1,20]
Livermore-22 \(\exp(-0.5x^2)\) 1 U[-1,1,20]