The algorithms without shape constraints produce extreme predictions when extrapolating (Figures 2a, 2c, 2e). For instance, many of the functions have poles at a temperature T close to zero which are visible in the plots as vertical lines. GP and GPC without shape constraints produced a few solutions which are wildly fluctuating over p even within the interpolation range. Within the interpolation range ITEA produced the best SR solutions for the Friction data sets (see Figure 2e as well as Table 4). However, the models do not conform to prior knowledge as we would expect that μstat decreases with increasing pressure and temperature and the models show a slight increase in μstat when increasing p⁠.
Figure 1:
Constraint violation frequency for the solutions of every algorithm over the 30 executions for the FDE data sets. Only for the Fuel Flow problem a feasible solution can be identified even without shape constraints. FI-2POP always produces feasible solutions due to the nature of its constraint handling mechanism.
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Constraint violation frequency for the solutions of every algorithm over the 30 executions for the FDE data sets. Only for the Fuel Flow problem a feasible solution can be identified even without shape constraints. FI-2POP always produces feasible solutions due to the nature of its constraint handling mechanism.

Figure 1:
Constraint violation frequency for the solutions of every algorithm over the 30 executions for the FDE data sets. Only for the Fuel Flow problem a feasible solution can be identified even without shape constraints. FI-2POP always produces feasible solutions due to the nature of its constraint handling mechanism.
View largeDownload slide

Constraint violation frequency for the solutions of every algorithm over the 30 executions for the FDE data sets. Only for the Fuel Flow problem a feasible solution can be identified even without shape constraints. FI-2POP always produces feasible solutions due to the nature of its constraint handling mechanism.

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Table 4:

Median NMSE values for the test data. Values are multiplied by 100 (percentage) and truncated at the second decimal place.

w/o. infow. info
GPGPCITEAAMLGPGPCFIITSCPR
Friction μdyn 8.28 7.73 4.35 12.99 12.53 16.30 35.90 8.07 
Friction μstat 7.22 5.44 4.46 6.82 9.98 7.76 11.83 1.77 
Flow stress 8.36 4.66 – 0.15 34.05 26.04 68.16 19.46 
Cars 75.18 76.23 75.06 74.72 76.86 77.67 76.64 73.83 
without noise         
Aircraft lift 0.63 0.15 0.22 0.00 0.80 1.01 0.14 0.00 
Flow psi 0.75 0.13 0.05 0.00 4.80 5.36 2.91 0.00 
Fuel flow 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Jackson 2.11 0.00 0.00 0.00 0.62 0.00 0.00 0.00 0.90 
Wave Power 14.55 30.82 2.26 2.74 18.71 80.34 21.31 13.50 
I.6.20 0.46 0.00 0.31 0.00 1.61 0.42 3.20 0.01 
I.9.18 2.88 2.49 0.91 4.98 4.03 16.16 0.74 1.20 
I.15.3x 0.34 0.01 0.01 0.02 0.36 0.04 0.01 0.01 
I.15.3t 0.21 0.01 0.00 0.00 0.15 0.03 0.00 0.00 
I.30.5 0.00 0.00 0.00 0.18 0.00 0.00 0.00 0.00 
I.32.17 0.76 1.13 8.07 42.36 2.07 12.76 2.42 7.79 
I.41.16 2.78 2.29 1.08 15.14 8.99 17.72 5.15 1.56 
I.48.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
II.6.15a 3.55 2.50 4.66 16.17 4.67 7.30 32.12 1.01 
II.11.27 0.00 0.00 0.00 0.00 0.00 0.07 0.00 0.00 
II.11.28 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
II.35.21 3.29 1.18 2.61 1.40 3.67 6.10 3.22 1.34 
III.9.52 116.25 20.44 66.16 19.88 104.56 106.48 71.93 33.41 
III.10.19 0.41 0.04 0.17 0.01 0.55 0.33 0.31 0.00 
with noise         
Aircraft lift 0.45 0.26 0.25 0.32 1.24 1.30 0.28 0.46 
Flow psi 0.75 0.29 0.21 0.37 5.90 6.02 4.65 0.57 
Fuel flow 0.21 0.24 0.18 0.34 0.30 0.30 0.25 0.25 
Jackson 2.11 0.28 0.31 0.38 3.18 0.24 0.25 0.30 0.83 
Wave Power 21.23 51.36 99.88 44.83 22.36 68.96 21.39 11.88 
I.6.20 1.09 0.40 0.56 0.45 2.14 0.78 3.61 0.55 
I.9.18 3.77 3.55 1.56 4.02 5.25 15.70 1.33 1.62 
I.15.3x 0.55 0.36 0.38 0.37 0.56 0.35 0.36 0.42 
I.15.3t 0.65 0.48 0.58 0.53 0.59 0.51 0.48 0.45 
I.30.5 0.34 0.35 0.62 0.81 0.32 0.33 0.34 0.39 
I.32.17 0.78 3.14 8.50 47.60 3.95 14.02 2.53 6.22 
I.41.16 3.13 2.32 3.47 15.19 6.68 19.72 5.05 2.93 
I.48.20 0.37 0.36 0.51 0.35 0.32 0.32 0.34 0.32 
II.6.15a 3.08 2.88 7.56 19.29 3.87 6.05 45.32 1.76 
II.11.27 0.35 0.39 1.06 0.62 0.37 0.61 0.41 0.47 
II.11.28 0.38 0.44 0.39 0.51 0.27 0.29 0.38 0.30 
II.35.21 3.88 1.33 2.43 2.10 4.38 7.49 4.27 1.34 
III.9.52 126.84 18.91 74.08 24.81 106.56 90.18 73.44 32.69 
III.10.19 0.85 0.38 0.70 0.64 0.91 0.64 0.70 0.46 
w/o. infow. info
GPGPCITEAAMLGPGPCFIITSCPR
Friction μdyn 8.28 7.73 4.35 12.99 12.53 16.30 35.90 8.07 
Friction μstat 7.22 5.44 4.46 6.82 9.98 7.76 11.83 1.77 
Flow stress 8.36 4.66 – 0.15 34.05 26.04 68.16 19.46 
Cars 75.18 76.23 75.06 74.72 76.86 77.67 76.64 73.83 
without noise         
Aircraft lift 0.63 0.15 0.22 0.00 0.80 1.01 0.14 0.00 
Flow psi 0.75 0.13 0.05 0.00 4.80 5.36 2.91 0.00 
Fuel flow 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
Jackson 2.11 0.00 0.00 0.00 0.62 0.00 0.00 0.00 0.90 
Wave Power 14.55 30.82 2.26 2.74 18.71 80.34 21.31 13.50 
I.6.20 0.46 0.00 0.31 0.00 1.61 0.42 3.20 0.01 
I.9.18 2.88 2.49 0.91 4.98 4.03 16.16 0.74 1.20 
I.15.3x 0.34 0.01 0.01 0.02 0.36 0.04 0.01 0.01 
I.15.3t 0.21 0.01 0.00 0.00 0.15 0.03 0.00 0.00 
I.30.5 0.00 0.00 0.00 0.18 0.00 0.00 0.00 0.00 
I.32.17 0.76 1.13 8.07 42.36 2.07 12.76 2.42 7.79 
I.41.16 2.78 2.29 1.08 15.14 8.99 17.72 5.15 1.56 
I.48.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
II.6.15a 3.55 2.50 4.66 16.17 4.67 7.30 32.12 1.01 
II.11.27 0.00 0.00 0.00 0.00 0.00 0.07 0.00 0.00 
II.11.28 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
II.35.21 3.29 1.18 2.61 1.40 3.67 6.10 3.22 1.34 
III.9.52 116.25 20.44 66.16 19.88 104.56 106.48 71.93 33.41 
III.10.19 0.41 0.04 0.17 0.01 0.55 0.33 0.31 0.00 
with noise         
Aircraft lift 0.45 0.26 0.25 0.32 1.24 1.30 0.28 0.46 
Flow psi 0.75 0.29 0.21 0.37 5.90 6.02 4.65 0.57 
Fuel flow 0.21 0.24 0.18 0.34 0.30 0.30 0.25 0.25 
Jackson 2.11 0.28 0.31 0.38 3.18 0.24 0.25 0.30 0.83 
Wave Power 21.23 51.36 99.88 44.83 22.36 68.96 21.39 11.88 
I.6.20 1.09 0.40 0.56 0.45 2.14 0.78 3.61 0.55 
I.9.18 3.77 3.55 1.56 4.02 5.25 15.70 1.33 1.62 
I.15.3x 0.55 0.36 0.38 0.37 0.56 0.35 0.36 0.42 
I.15.3t 0.65 0.48 0.58 0.53 0.59 0.51 0.48 0.45 
I.30.5 0.34 0.35 0.62 0.81 0.32 0.33 0.34 0.39 
I.32.17 0.78 3.14 8.50 47.60 3.95 14.02 2.53 6.22 
I.41.16 3.13 2.32 3.47 15.19 6.68 19.72 5.05 2.93 
I.48.20 0.37 0.36 0.51 0.35 0.32 0.32 0.34 0.32 
II.6.15a 3.08 2.88 7.56 19.29 3.87 6.05 45.32 1.76 
II.11.27 0.35 0.39 1.06 0.62 0.37 0.61 0.41 0.47 
II.11.28 0.38 0.44 0.39 0.51 0.27 0.29 0.38 0.30 
II.35.21 3.88 1.33 2.43 2.10 4.38 7.49 4.27 1.34 
III.9.52 126.84 18.91 74.08 24.81 106.56 90.18 73.44 32.69 
III.10.19 0.85 0.38 0.70 0.64 0.91 0.64 0.70 0.46 
View Large
Figure 2:
Partial dependence plots for the Friction μstat models found by each algorithm over the 30 runs. Dashed lines mark the subspace from which training and test points were sampled. Algorithms with shape constraints (b, d, f) produce SR solutions which conform to prior knowledge and have better extrapolation behavior but increased prediction error (cf. Table 4).
View largeDownload slide

Partial dependence plots for the Friction μstat models found by each algorithm over the 30 runs. Dashed lines mark the subspace from which training and test points were sampled. Algorithms with shape constraints (b, d, f) produce SR solutions which conform to prior knowledge and have better extrapolation behavior but increased prediction error (cf. Table 4).

Figure 2:
Partial dependence plots for the Friction μstat models found by each algorithm over the 30 runs. Dashed lines mark the subspace from which training and test points were sampled. Algorithms with shape constraints (b, d, f) produce SR solutions which conform to prior knowledge and have better extrapolation behavior but increased prediction error (cf. Table 4).
View largeDownload slide

Partial dependence plots for the Friction μstat models found by each algorithm over the 30 runs. Dashed lines mark the subspace from which training and test points were sampled. Algorithms with shape constraints (b, d, f) produce SR solutions which conform to prior knowledge and have better extrapolation behavior but increased prediction error (cf. Table 4).

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