All constraints for FDE and FYE have been determined by analyzing the known expression and its partial derivatives. The formulas for the FDE and FYE problem instances are shown in Table 1. For all these problem instances we have generated data sets from the known formula by randomly sampling the input space using 100 data points for training and 100 data points for testing. For each of the data sets we generated two versions: one without noise and one where we have added normally distributed noise to the target variable $y'=y+N(0,0.05σy)$. Accordingly, the optimally achievable NMSE for the noisy problem instances is $0.25%$.

Table 1:

Synthetic problem instances used for testing. The first three functions are from the FDE data sets, the rest from the FEY data sets.

NameFormula
Aircraft lift $CL=CLα(α+α0)+CLδeδeSHTSref$
Flow psi $Ψ=V∞rsin(θ2π)1-Rr2+Γ2πlogrR$
Fuel flow $m˙=p0A★T0γR21+γ(γ+1)/(γ-1)$
Jackson 2.11 $q4πεy24πεVoltd-qdy3y2-d22$
Wave power $-325G4c5m1m22m1+m2r5$
I.6.20 $exp-θσ2212πσ$
I.9.18 $Gm1m2x2-x12+y2-y12+z2-z12$
I.15.3x $x-ut1-u2c2$
I.15.3t $t-uxc211-u2c2$
I.30.5 $asinlambdnd$
I.32.17 $12εcEf28πr23ω4ω2-ω022$
I.41.16 $hω3π2c2exphωkbT-1$
I.48.20 $mc21-v2c2$
II.6.15a $pd4πε3zr5x2+y2$
II.11.27 $nα1-nα3εEf$
II.11.28 $1+nα1-nα3$
II.35.21 $nrhomomtanhmomBkbT$
III.9.52 $pdEfthsinω-ω0t221ω-ω0t22$
III.10.19 $momBx2+By2+Bz2$
NameFormula
Aircraft lift $CL=CLα(α+α0)+CLδeδeSHTSref$
Flow psi $Ψ=V∞rsin(θ2π)1-Rr2+Γ2πlogrR$
Fuel flow $m˙=p0A★T0γR21+γ(γ+1)/(γ-1)$
Jackson 2.11 $q4πεy24πεVoltd-qdy3y2-d22$
Wave power $-325G4c5m1m22m1+m2r5$
I.6.20 $exp-θσ2212πσ$
I.9.18 $Gm1m2x2-x12+y2-y12+z2-z12$
I.15.3x $x-ut1-u2c2$
I.15.3t $t-uxc211-u2c2$
I.30.5 $asinlambdnd$
I.32.17 $12εcEf28πr23ω4ω2-ω022$
I.41.16 $hω3π2c2exphωkbT-1$
I.48.20 $mc21-v2c2$
II.6.15a $pd4πε3zr5x2+y2$
II.11.27 $nα1-nα3εEf$
II.11.28 $1+nα1-nα3$
II.35.21 $nrhomomtanhmomBkbT$
III.9.52 $pdEfthsinω-ω0t221ω-ω0t22$
III.10.19 $momBx2+By2+Bz2$

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