Article Contents

2023, Volume 16, Issue 2: 220-239. Doi: 10.3934/dcdss.2022198

This issue Previous Article Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations Next Article Comparison of simulation-based algorithms for parameter estimation and state reconstruction in nonlinear state-space models

Downscaling shallow water simulations using artificial neural networks and boosted trees

1.
LEMON, Inria, France
2.
IMAG, Univ. Montpellier, CNRS, Montpellier, France
3.
HSM, Univ. Montpellier, CNRS, IRD, Montpellier, France

^*Corresponding author: Antoine Rousseau
^*Corresponding author: Antoine Rousseau

Received: October 2022

Revised: December 2022

Early access: December 2022

Published: February 2023

Abstract / Introduction Full Text(HTML) Figure(7) / Table(20) Related Papers Cited by

Abstract

We present the application of two statistical artificial intelligence tools for multi-scale shallow water simulations. Artificial neural networks (ANNs) and boosted trees (BTs) are used to model the relationship between low-resolution (LR) and high-resolution (HR) information derived from simulations provided in the learning phase. The two statistical models are analyzed (and compared) through hyper-parameters such as the number of epochs and the network structure for ANNs, and the learning rate, tree depth and number for BTs. This analysis is performed through 4 numerical experiments the input datasets of which (for the learning, validation and test phases) are varied through the boundary conditions of the flow numerical simulation.

The performance of the ANNs is remarkably consistent, regardless of the choice made for the training/validation/testing set. The performance improves with the number of epochs and the number of neurons. For a given number of neurons, a single-layer structure performs better than multi-layer structures. BTs perform significantly better than ANNs in 2 experiments, with an error 10 to 100 times lower and a computational cost 5 to 10 times larger). However, when the validation datasets differ from the training datasets, the performance of BTs performance is strongly degraded, with a modelling error more than one order of magnitude larger than that of ANNs.

Used in conjunction with upscaled flood models such as porosity models, these techniques appear as a promising operational alternative to direct flood hazard assessment from HR flow simulations.

Keywords:

Mathematics Subject Classification: Primary: 86A05, 86A32; Secondary: 68T07.

Citation:

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Figure 1. Upscaling and downscaling. Definition sketch

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Figure 2. Definition sketch for shallow water models structure and variables. The red and blue lines represent respectively the (steady) bottom elevation and the (unsteady) free surface elevation

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Figure 3. Definition sketch for the experiment plan. Top : representation in the $ (x, h) $ plane for a given time $ t $. Bottom : water depth contour lines in the $ (x, t) $ plane. Owing to solution self-similarity, the speeds of Points A and B are constant, hence the straight, $ h $-contour lines in the $ (x, t) $ plane (bottom)

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Figure 4. Experiment 1 - ANN and BT best Mean Squared Error (MSE) as a function of the Upscaling Ratio (UR) for the various reconstructed variables

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Figure 5. Experiment 2 - ANN and BT best Mean Squared Error (MSE) as a function of the Upscaling Ratio (UR) for the various reconstructed variables

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Figure 6. Experiment 3 -ANN and BT best Mean Squared Error (MSE) as a function of the Upscaling Ratio (UR) for the various reconstructed variables

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Figure 7. Experiment 4 - ANN and BT best Mean Squared Error (MSE) as a function of the Upscaling Ratio (UR) for the various reconstructed variables

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Table 1. BVPs : model parameters

Symbol	Meaning	Numerical value
$ L $	Domain length	100 m
$ \Delta x_{HR} $	HR cell size	0.125 m
$ \Delta x_{LR} $	LR cell size	0.625 m, 5 m, 10 m

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Table 2. ANN hyperparameters. The number of epochs is the number of times the entire data set is used in the training process

Hyperparameters	Numerical values
Number of epochs	50,150 or 500
Batch size	32
Number of neurons (1-layer configuration)	100 or 500
Number of neurons (2-layer configuration)	(50, 50) or (100,100)
Number of neurons (3-layer configuration)	(50, 50, 50) or (75, 75, 75)

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Table 3. BT hyperparameters

Hyperparameter	Numerical values
Learning rate	0.1
Maximum depth	2 or 4
Minimum samples per leaf	1 sample, 2% of set
Number of trees	7, 20 or 50
Subsample share used for each tree	50%

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Table 4. Experiment plan. $ h_0 = $ 1 m for all simulations

Experiment	Training		Validation		Test
Experiment	$ h_1 $ (m)	sample size	$ h_1 $ (m)	sample size	$ h_1 $ (m)	sample size
1	$ {0.7 , 0.9 } $	880	$ {0.7 , 0.9 } $	222	$ 0.8 $	551
2	$ {0.7 , 0.9 } $	1102	$ {0.75 , 0.85 } $	1102	$ 0.8 $	551
3	$ {0.7 , 0.8 } $	880	$ {0.7 , 0.8 } $	222	$ 0.9 $	551
4	$ {0.7 , 0.85} $	1102	$ {0.75 , 0.8 } $	1102	$ 0.9 $	551

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Table 5. Experiment 1 - ANN best performance. UR: Upscaling Ratio. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, [b_i]] $, with $ a $ the number of epochs, $ b_i $ the number of neurons in Layer $ i $

UR	Best performance	Reconstructed
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 1.2 \times 10^{-5} $	$ 4.4\times 10^{-5} $	$ 1.7\times 10^{-5} $
	CPU time (s)	(170, 0.38)	(83, 0.23)	(63, 0.24)
	Param	[500, [100]]	[500, [100]]	[500, [100]]
20	MSE (m$ ^2 $)	$ 1.7\times 10^{-5} $	$ 3.9\times 10^{-5} $	$ 1.3\times 10^{-5} $
	CPU time (s)	(140, 0.15)	(83, 0.23)	(66, 0.27)
	Param	[500, [100,100]]	[500, [100]]	[500, [100]]
80	MSE (m$ ^2 $)	$ 7.4\times 10^{-5} $	$ 7.5\times 10^{-5} $	$ 5.5\times 10^{-5} $
	CPU time	(57, 0.14)	(130, 0.067)	(69, 0.32)
	Param	[500, [75, 75, 75]]	[500, [500]]	[500, [100,100]]

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Table 6. Experiment 1 - BT best performance. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, b, c] $, with $ a $ the maximum depth, $ b $ the minimum fraction of samples per leaf, $ c $ the number of trees

UR	Best performance	Reconstructed variable
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 2.1\times 10^{-6} $	$ 3.9\times 10^{-7} $	$ 4.3e\times 10^{-7} $
	CPU time (s)	($ 1.8\times 10^{3} $, $ 8.8\times 10^{-1}) $	($ 3.8e\times 10^{2} $, $ 3.6\times 10^{-1}) $	($ 4.2\times 10^{2} $, $ 3.7\times 10^{-1} $)
	Param	[4, 1.0, 50]	[4, 1.0, 50]	[4, 1.0, 50]
20	MSE (m$ ^2 $)	$ 4.\times 10^{-6} $	$ 1.\times 10^{-6} $	$ 8.\times 10^{-7} $
	CPU time (s)	($ 4.4\times 10^{2} $, $ 6.7\times 10^{-1} $)	($ 1.3\times 10^{2} $, $ 3.1\times 10^{-1} $)	($ 1.3\times 10^{2} $, $ 3.1\times 10^{-1} $)
	Param	[4, 1.0, 50]	[4, 1.0, 50]	[4, 1.0, 50]
80	MSE (m$ ^2 $)	$ 1.1\times 10^{-5} $	$ 1.1\times 10^{-5} $	$ 9.4\times 10^{-6} $
	CPU time (s)	($ 1.5\times 10^{2} $, $ 6.3\times 10^{-1} $)	($ 3.8e\times 10^{1} $, $ 2.9\times 10^{-1} $)	($ 4.5\times 10^{1} $, $ 3.8\times 10^{-1} $)
	Param	[4, 1.0, 50]	[4, 0.02, 50]	[2, 1.0, 50]

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Table 7. Experiment 1 - ANN performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter	Value	MSE (m$ ^2 $)	Training time (s)	Evaluation time (ms)
Epochs	50	$ 3.0\times 10^{-4} $	10.9	0.23
	150	$ 7.2\times 10^{-5} $	24.7	0.24
	500	$ 3.0\times 10^{-5} $	80.6	0.16
Layer structure	[100]	$ 1.1\times 10^{-4} $	31.3	0.20
	[500]	$ 6.8\times 10^{-5} $	64.1	0.18
	[50, 50]	$ 2.0\times 10^{-4} $	38.7	0.25
	[100,100]	$ 1.1\times 10^{-4} $	32.9	0.16
	[50, 50, 50]	$ 2.0\times 10^{-4} $	38.2	0.23
	[75, 75, 75]	$ 1.1\times 10^{-4} $	27.1	0.22

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Table 8. Experiment 1 - BT performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter	Value	MSE (m$ ^2 $)	Training time	Evaluation time
Maximum depth	2	$ 4.9\times 10^{-4} $	47.4	0.33
Maximum depth	4	$ 4.7\times 10^{-4} $	60.5	0.30
Minimum samples per leaf	1	$ 4.7\times 10^{-4} $	58.8	0.33
Minimum samples per leaf	2%	$ 4.8\times 10^{-4} $	49.2	0.31
Number of trees	7	$ 1.3\times 10^{-3} $	17.3	0.30
	20	$ 9.2\times 10^{-5} $	43.3	0.31
	50	$ 2.6\times 10^{-6} $	101	0.35

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Table 9. Experiment 2 - ANN best performance. UR: Upscaling Ratio. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, [b_i]] $, with $ a $ the number of epochs, $ b_i $ the number of neurons in Layer $ i $

UR	Best performance	Reconstructed variable
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 5.7\times 10^{-5} $	$ 1.8\times 10^{-5} $	$ 3.8\times 10^{-6} $
	CPU time (s)	($ 3.6\times 10^{1} $, $ 4.0\times 10^{-1} $)	($ 1.8e\times 10^{2} $, $ 1.3\times 10^{-1} $)	($ 1.8\times 10^{2} $, $ 1.3\times 10^{-1} $)
	Param	[150, [100]]	[500, [500]]	[500, [500]]
20	MSE (m$ ^2 $)	$ 2.\times 10^{-5} $	$ 1.7\times 10^{-5} $	$ 6.7\times 10^{-6} $
	CPU time (s)	($ 1.1\times 10^{2} $, $ 4.4\times 10^{-1} $)	($ 6.9\times 10^{+1} $, $ 4.9\times 10^{-1} $)	($ 7.9\times 10^{1} $, $ 2.7\times 10^{-1} $)
	Param	[500, [100]]	[500, [100]]	[500, [100]]
80	MSE (m$ ^2 $)	$ 1.8\times 10^{-4} $	$ 6.7\times 10^{-5} $	$ 4.1\times 10^{-5} $
	CPU time (s)	($ 1.4\times 10^{2} $, $ 3.9\times 10^{-1} $)	($ 6.6\times 10^{1} $, $ 1.9\times 10^{-1} $)	($ 6.4\times 10^{1} $, $ 1.7\times 10^{-1} $)
	Param	[500, [100]]	[500, [75, 75, 75]]	[500, [75, 75, 75]]

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Table 10. Experiment 2 - BT best performance. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, b, c] $, with $ a $ the maximum depth, $ b $ the minimum fraction of samples per leaf, $ c $ the number of trees

UR	Best performance	Reconstructed variable
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 6.5\times 10^{-3} $	$ 1.6\times 10^{-3} $	$ 1.6\times 10^{-3} $
	CPU time	($ 2.3\times 10^{3} $, 3.6)	($ 5.3\times 10^{2} $, 1.2)	($ 5.3\times 10^{2} $, 1.2)
	Param	[4, 1.0, 50]	[4, 1.0, 50]	[4, 1.0, 50]
20	MSE (m$ ^2 $)	$ 6.0\times 10^{-3} $	$ 1.3\times 10^{-3} $	$ 1.4\times 10^{-3} $
	CPU time	($ 5.5\times 10^{2} $, 2.0)	($ 1.9\times 10^{2} $, $ 8.9\times 10^{-1} $)	$ (1.2\times 10^{2}, 8.4\times 10^{-1}) $
	Param	[4, 1.0, 50]	[4, 1.0, 50]	[4, 0.02, 50]
80	MSE (m$ ^2 $)	$ 5.5\times 10^{-3} $	$ 1.6\times 10^{-3} $	$ 1.6\times 10^{-3} $
	CPU time	$ (1.5\times 10^{2}, 1.6) $	$ (4.4\times 10^{1}, 7.8\times 10^{-1}) $	$ (5.2\times 10^{1}, 7.7\times 10^{-1}) $
	Param	[4, 0.02, 50]	[4, 0.02, 50]	[4, 0.02, 50]

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Table 11. Experiment 2 - ANN performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter	Value	MSE (m$ ^2 $)	Training time	Evaluation time
Epochs	50	$ 2.6\times 10^{-4} $	11.4	0.32
	150	$ 5.3\times 10^{-5} $	32.8	0.27
	500	$ 2.4\times 10^{-5} $	92.3	0.27
Layer Structure	[100]	$ 1.1\times 10^{-4} $	37.7	0.29
	[500]	$ 4.1\times 10^{-5} $	76.0	0.20
	[50, 50]	$ 1.4\times 10^{-4} $	44.7	0.31
	[100,100]	$ 1.2\times 10^{-4} $	46.6	0.38
	[50, 50, 50]	$ 1.6\times 10^{-4} $	33.5	0.22
	[75, 75, 75]	$ 1.1\times 10^{-4} $	34.5	0.31

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Table 12. Experiment 2 - BT performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter	Value	MSE (m$ ^2 $)	Training time	Evaluation time
Maximum depth	2	$ 2\times 10^{-3} $	57.6	0.67
Maximum depth	4	$ 2\times 10^{-3} $	75.2	0.67
Minimum samples per leaf	1	$ 2\times 10^{-3} $	72.4	0.70
Minimum samples per leaf	2%	$ 2\times 10^{-3} $	60.4	0.64
Number of trees	7	$ 2\times 10^{-3} $	21.3	0.49
	20	$ 2\times 10^{-3} $	53.6	0.62
	50	$ 2\times 10^{-3} $	124.3	0.89

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Table 13. Experiment 3 - ANN best performance. UR: Upscaling Ratio. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, [b_i]] $, with $ a $ the number of epochs, $ b_i $ the number of neurons in Layer $ i $

UR	Best performance	Reconstructed variable
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 1.5\times 10^{-5} $	$ 7.1\times 10^{-5} $	$ 1.1\times 10^{-5} $
	CPU time (s)	$ (1.4e\times 10^{2} 1.6\times 10^{-1}) $	$ (5.6\times 10^{1}, 1.3\times 10^{-1}) $	$ (6.2\times 10^{1}, 2.4\times 10^{-1}) $
	Param	[500, [50, 50, 50]]	[500, [75, 75, 75]]	[500, [100]]
20	MSE (m$ ^2 $)	$ 1.3\times 10^{-5} $	$ 2.6\times 10^{-5} $	$ 9.8\times 10^{-6} $
	CPU time (s)	$ (7.9\times 10^{1}, 1.7\times 10^{-1}) $	$ (5.5\times 10^{1}, 1.7\times 10^{-1}) $	$ (5.9\times 10^{1}, 3.2\times 10^{-1}) $
	Param	[500, [75, 75, 75]]	[500, [75, 75, 75]]	[500, [100]]
80	MSE (m$ ^2 $)	$ 3.76\times 10^{-5} $	$ 5.99\times 10^{-5} $	$ 4.45\times 10^{-5} $
	CPU time (s)	$ (8.0\times 10^{1}, 1.6\times 10^{-1}) $	$ (8.3\times 10^{1}, 3.1\times 10^{-1}) $	$ (8.3\times 10^{1}, 7.7\times 10^{-2}) $
	Param	[500, [50, 50, 50]]	[500, [100,100]]	[500, [75, 75, 75]]

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Table 14. Experiment 3 - BT best performance. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, b, c] $, with $ a $ the maximum depth, $ b $ the minimum fraction of samples per leaf, $ c $ the number of trees

UR	Best performance	Reconstructed variable
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 1.9\times 10^{-6} $	$ 4.9\times 10^{-7} $	$ 4.1\times 10^{-7} $
	CPU time (s)	$ (2.7\times 10^{3}, 1.5) $	$ (4.6\times 10^{2}, 3.7\times 10^{-1}) $	$ (3.9\times 10^{2}, 3.8\times 10^{-1}) $
	Param	[4, 1.0, 50]	[4, 1.0, 50]	[4, 1.0, 50]
20	MSE (m$ ^2 $)	$ 3.4\times 10^{-6} $	$ 7.8\times 10^{-7} $	$ 6.9\times 10^{-7} $
	CPU time (s)	$ (7.2\times 10^{2}, 1.3) $	$ (1.4\times 10^{2}, 3.2\times 10^{-1}) $	$ (1.4\times 10^{2}, 3.2\times 10^{-1}) $
	Param	[4, 1.0, 50]	[4, 1.0, 50]	[4, 1.0, 50]
80	MSE (m$ ^2 $)	$ 1.78\times 10^{-5} $	$ 7.34\times 10^{-6} $	$ 7.53\times 10^{-6} $
	CPU time (s)	$ (1.7\times 10^{2}, 6.7\times 10^{-1}) $	$ (4.5\times 10^{1}, 3.8\times 10^{-1}) $	$ (4.6\times 10^{1}, 3.8\times 10^{-1}) $
	Param	[4, 1.0, 50]	[2, 1.0, 50]	[2, 1.0, 50]

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Table 15. Experiment 3 - ANN performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter	Value	MSE (m$ ^2 $)	Training time	Evaluation time
Epochs	50	$ 3.4\times 10^{-4} $	10.3	0.21
	150	$ 7.5\times 10^{-5} $	28.1	0.31
	500	$ 1.5\times 10^{-5} $	71.5	0.20
Layers structure	[100]	$ 1.3\times 10^{-4} $	29.4	0.24
	[500]	$ 5.9\times 10^{-5} $	62.7	0.18
	[50, 50]	$ 1.771\times 10^{-4} $	37.5	0.22
	[100,100]	$ 1.3\times 10^{-4} $	31.3	0.26
	[50, 50, 50]	$ 2.1\times 10^{-4} $	28.2	0.31
	[75, 75, 75]	$ 1.5\times 10^{-4} $	30.7	0.23

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Table 16. Experiment 3 - BT performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter	Value	MSE (m$ ^2 $)	Training time	Evaluation time
Maximum depth	2	$ 4.8\times 10^{-4} $	50.1	0.34
Maximum depth	4	$ 4.6\times 10^{-4} $	66.1	0.32
Minimum samples per leaf	1	$ 4.7\times 10^{-4} $	63.1	0.33
Minimum samples per leaf	2%	$ 4.8\times 10^{-4} $	53.1	0.33
Number of trees	7	$ 1.3\times 10^{-3} $	18.2	0.30
	20	$ 9.3\times 10^{-5} $	50.1	0.34
	50	$ 3.8\times 10^{-6} $	106	0.36

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Table 17. Experiment 4 - ANN best performance. UR: Upscaling Ratio. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, [b_i]] $, with $ a $ the number of epochs, $ b_i $ the number of neurons in Layer $ i $

UR	Best performance	Reconstructed variable
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 9.3\times 10^{-5} $	$ 8.4\times 10^{-5} $	$ 1.8\times 10^{-5} $
	CPU time (s)	$ (1.4\times 10^{2}, 1.0) $	$ (1.4\times 10^{2}, 8.1\times 10^{-1}) $	$ (1.0\times 10^{2}, 9.4\times 10^{-1}) $
	Param	[150, [500]]	[500, [100, 100]]	[500, [100]]
20	MSE (m$ ^2 $)	$ 7.6\times 10^{-5} $	$ 5.6\times 10^{-5} $	$ 2.1\times 10^{-5} $
	CPU time (s)	$ (3.3\times 10^{2}, 1.6\times 10^{-1}) $	$ (1.9\times 10^{2}, 2.2\times 10^{-1}) $	$ (9.9\times 10^{^1}, 6.1\times 10^{-1}) $
	Param	[500, [500]]	[500, [500]]	[500, [50, 50]]
80	MSE (m$ ^2 $)	$ 2.8\times 10^{-4} $	$ 1.0\times 10^{-4} $	$ 1.0\times 10^{-4} $
	CPU time (s)	$ (1.1\times 10^{2}, 4.9\times 10^{-1}) $	$ (9.2\times 10^{1}, 4.4\times 10^{-1}) $	$ (5.1\times 10^{1}, 1.2\times 10^{-1}) $
	Param	[500, [100]]	[500, [100]]	[500, [75, 75, 75]]

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Table 18. Experiment 4 - BT best performance. CPU time format: $ (T_1, T_2) $, with $ T_1 $ the training time, $ T_2 $ the evaluation time. Hyperparameter format (Param): $ [a, b, c] $, with $ a $ the maximum depth, $ b $ the minimum fraction of samples per leaf, $ c $ the number of trees

UR	Best performance	Reconstructed variable
UR	Best performance	$ (h, q) $	$ \sqrt{h} $	$ h $
5	MSE (m$ ^2 $)	$ 4.4\times 10^{-3} $	$ 1.6\times 10^{-3} $	$ 1.6\times 10^{-3} $
	CPU time (s)	$ (2.0\times 10^{3}, 6.0) $	$ (4.6\times 10^{2}, 1.7) $	$ (4.2\times 10^{2}, 1.5) $
	Param	[2, 1.0, 50]	[2, 0.02, 50]	[2, 0.02, 50]
20	MSE (m$ ^2 $)	$ 4.4\times 10^{-3} $	$ 1.5\times 10^{-3} $	$ 1.6\times 10^{-3} $
	CPU time (s)	$ (5.6\times 10^{2}, 3.1) $	$ (1.5\times 10^{2}, 1.2) $	$ (1.3\times 10^{2}, 1.0) $
	Param	[2, 1.0, 50]	[2, 1.0, 50]	[2, 0.02, 50]
80	MSE (m$ ^2 $)	$ 4.4\times 10^{-3} $	$ 1.6\times 10^{-3} $	$ 1.6\times 10^{-3} $
	CPU time (s)	$ (2.0\times 10^{2}, 1.9) $	$ (6.8\times 10^{1}, 8.9\times 10^{-1}) $	$ (5.8\times 10^{1}, 9.5\times 10^{-1}) $
	Param	[2, 0.02, 50]	[2, 1.0, 50]	[2, 0.02, 50]

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Table 19. Experiment 4 - ANN performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter	Value	MSE (m$ ^2 $)	Training time	Evaluation time
Epochs	50	$ 5.8\times 10^{-4} $	14.1	0.61
	150	$ 1.3\times 10^{-4} $	41.5	0.61
	500	$ 2.9\times 10^{-5} $	118	0.32
Layers structure	[100]	$ 2.2\times 10^{-4} $	50.7	0.59
	[500]	$ 7.8\times 10^{-5} $	100	0.45
	[50, 50]	$ 3.2\times 10^{-4} $	47.4	0.51
	[100, 100]	$ 2.5\times 10^{-4} $	64.0	0.52
	[50, 50, 50]	$ 3.9\times 10^{-4} $	42.3	0.59
	[75, 75, 75]	$ 2.2\times 10^{-4} $	42.6	0.44

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Table 20. Experiment 4 - BT performance for various hyperparameter sets. Reconstructed variable: $ h $. UR: 20

hyperparameter		MSE (m$ ^2 $)	Training time	Evaluation time
Maximum depth	2	$ 2\times 10^{-3} $	66.4	0.76
Maximum depth	4	$ 2\times 10^{-3} $	87.2	0.79
Minimum samples per leaf	1	$ 2\times 10^{-3} $	82.4	0.79
Minimum samples per leaf	2%	$ 2\times 10^{-3} $	71.2	0.76
Number of trees	7.00	$ 4\times 10^{-3} $	24.3	0.52
	20	$ 2\times 10^{-3} $	64.1	0.78
	50	$ 2\times 10^{-3} $	142	1.02

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