Table 2 Comparison of Active Subspaces, PyDimension, BuckiNet, and BSM across validation and application cases: Rayleigh problem, Colebrook equation, Malkus waterwheel, Rayleight-Bénard convection, Blasius boundary layer, Velocity scaling, Wall flux, Skin friction, MHD generator, Laser-metal interaction

Method	Active	PyDim-	Bucki-	BSM	IT-π
	Subspaces	ension	Net		(Current)
Rayleigh problem	✓	✓	✓	✓	✓
Colebrook equation	✓	×	✓	×	✓
Malkus waterwheel	×	×	×	×	✓
Rayleight-Bénard convection	×	✓	×	×	✓
Blasius boundary layer	✓	✓	✓	✓	✓
Velocity scaling	72%	21%	50%	N/A	12%
Wall shear stress	78%	54%	74%	N/A	19%
Wall heat flux	62%	44%	37%	N/A	9%
Skin friction	19%	27%	75%	N/A	17%
MHD generator	7%	7%	98%	N/A	5%
Laser-metal interaction	94%	25%	100%	N/A	23%

For the validation cases, the table presents whether the methods could identify the correct dimensionless variables for validations cases (✓ or × ). For the application cases, the table shows the normalized irreducible error \({\tilde{\epsilon }}_{LB}\) associated to the dimensionless input and output variables identified by each method with lower values indicating better performance.

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