# Branin-Hoo

In this case the true underlying function HyGP had to approximate is the 2D Branin-Hoo function (Viana and Haftka 2009):

$f(z_{1}, z_{2})=\left( z_{2} - 5.1\dfrac{z_{1}^2}{4\pi^2} + 5\dfrac{z_{1}}{\pi} - 6\right)^{2} + 10\left(1 - \dfrac{\pi}{8}\right) cos(z_{1}) + 10$

BUILDING DATA SET:
30-point Optimal Latin Hypercube DoE in [-5, 10] x [0, 15]
Available here: braninhoo_input_file

VALIDATION DATA SET:
1369-point Full Factorial DoE  [-6.5 : 0.5 : 11.5] x [-1.5 : 0.5 : 16.5]
Available here: braninhoo_test_dataset

HyGP hyperparameters (see braninhoo_input_file):
Population size: 200
Generations: 50
Primitives: +, -, *, / (protected), ^2, ^3, sin, cos, exp

Results:
The experiment done with 10 random initial guesses for numerical coefficients (10-guesses approach) returned as best model the following symbolic expression:

$\tilde{f}(z_{1},z_{2}) = 9.60209703926\, cos \left(1.00000053842\, \mathrm{z_{1}}\right) - 12.0000255964\, \mathrm{z_{2}} - 19.0985885529\, \mathrm{z_{1}} + 3.1830982295\, \mathrm{z_{1}}\, \mathrm{z_{2}} - 0.258368814492\, {\mathrm{z_{1}}}^2\, \mathrm{z_{2}} + 4.08324072854\, {\mathrm{z_{1}}}^2 - 0.411206971382\, {\mathrm{z_{1}}}^3 + 0.0166886394138\, {\mathrm{z_{1}}}^4 + 1.00000121669\, {\mathrm{z_{2}}}^2 + 46.0001253703$

characterised by a coefficient of determination R2=1 on the validation data. The returned expression $\tilde{f}$ is as a matter of fact the original underlying Branin-Hoo function, if minor errors on the numerical coefficients are neglected.

In the video below the evolution leading to the correct identification of the Branin-Hoo function is shown:

References:

• F.A.C. Viana and R.T. Haftka. Cross validation can estimate how well prediction variance correlates with error. AIAA Journal, 47(9):2266–2270, 2009.
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