Group sparse Bayesian learning for data-driven discovery of explicit model forms with multiple parametric datasets

Luning Sun; Pan Du; Hao Sun; Jian-Xun Wang

doi:10.3934/naco.2022040

Article Contents

2024, Volume 14, Issue 1: 190-213. Doi: 10.3934/naco.2022040

This issue Previous Article The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems Next Article Traditional and fuzzy goal programming approaches for resource dependent assembly line balancing problem

Group sparse Bayesian learning for data-driven discovery of explicit model forms with multiple parametric datasets

1.
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA
2.
Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China

^*Corresponding author: Jian-Xun Wang
^*Corresponding author: Jian-Xun Wang

The paper is handled by Andreas Mang as the guest editor.

Received: February 2022

Revised: November 2022

Early access: December 2022

Published: March 2024

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Abstract

Extracting the explicit governing equations of a dynamic system from limited data has attracted increasing attention in the data-driven modeling community. Compared to black-box learning approaches, the sparse-regression-based learning method enables discovering an analytical model form from data, which is more appealing due to its white-box nature. However, distilling explicit equations from real-world measurements with data uncertainty is challenging, where many existing methods are less robust. Moreover, it is unclear how to efficiently learn a parametric system from multiple data sets with different parameters. This paper presents a group sparse Bayesian learning approaches to uncover the explicit model forms of a parametric dynamical system with estimated uncertainties. A deep neural network is constructed to improve the calculation of derivatives from noisy measurements. Group sparsity is leveraged to enable synchronous learning from a group of parametric datasets governed by the equations with the same functional form but different parameter settings. The proposed approach has been studied over a few linear/nonlinear ODE systems in explicit and implicit settings. In particular, a simplified parametric model of intracranial dynamics was identified from multiple synthetic datasets with different patient-specific parameters. The numerical results demonstrated the effectiveness of the proposed approach and the merit of synchronous learning from multiple datasets in a group sparsifying Bayesian setting.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Schematics of the Bayesian group-sparsifying equation discovery framework with (a) neural network pre-processing black: noisy signal. blue: denoised signal. red: estimated noises.; (b) SINDy-type library construction; (c) group sparse Bayesian learning; (d) and example for identified sparse structure and posterior

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Figure 2. Identified systems of two different set of parameters with 5% noise: (1st column) $ \boldsymbol{\lambda} = [-0.05, 1.5, -1.5, -0.05] $ and (2nd column) $ \boldsymbol{\lambda} = [-0.15, 2, -2, -0.15] $. Only the trajectory of the state variable $ x $ is shown and the propagated uncertainty is given by the 3-$ \sigma $ region. The 2nd row is the zoomed-in view

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Figure 3. Identified systems of two different sets of parameters with 5% noise: (1st column) $ \boldsymbol{\lambda} = [-0.05, 1.5, -1.5, -0.05] $ and (2nd column) $ \boldsymbol{\lambda} = [-0.15, 2, -2, -0.15] $. Only the trajectory of the state variable $ x $ is shown and the propagated uncertainty is given by the 3-$ \sigma $ region. The 2nd row is the zoomed-in view

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Figure 4. Identified systems of two different set of parameters with 2% noise: (1st column) $ [j_x, V_{max}, K_m] = [0.3, 1.2, 0.2] $ and (2nd column) $ [j_x, V_{max}, K_m] = [0.3, 1.8, 0.4] $. The trajectory of the state variable $ x $ is shown and the propagated uncertainty is given by the 3-$ \sigma $ region. The 2nd row is the zoomed-in view

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Figure 5. Identified systems of two different set of parameters with 1% noise: (1st column) $ [C_{an}, R_{f}, k_{E}] = [0.125, 2.38\times10^{3}, 0.231] $ and (2nd column) $ [C_{an}, R_{f}, k_{E}] = [0.15, 2.38\times10^{3}, 0.175] $. The trajectory of the ICP is shown and the propagated uncertainty is given by the 3-$ \sigma $ region

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Table 1. System identification results for the parametric linear systems, $ \frac{dx}{dt} = \lambda_1x + \lambda_2y $, state $ \frac{dy}{dt} = \lambda_3x + \lambda_4y $. ($ 5\% $ data noise), where $ \lambda_{i} $ denotes the relevant coefficient, $ C(\text{const1}) $ denotes the redundant constant coefficient and $ \sigma(C(\text{const1})) $ denotes the standard deviation for equation $ \frac{dx}{dt} = \lambda_1x + \lambda_2y $, while $ C(\text{const2}) $ and $ \sigma(C(\text{const2})) $ are corresponding values for equation $ \frac{dy}{dt} = \lambda_3x + \lambda_4y $

Identified systems with group sparsity
Case	1	2	3	4	5	6	7	8	9
$ \lambda_1 $	-0.0493	-0.0494	-0.0495	-0.1004	-0.1007	-0.0999	-0.1499	-0.1490	-0.1504
$ \sigma(\lambda_1) $	0.0019	0.0019	0.0019	0.0017	0.0016	0.0016	0.0015	0.0015	0.0015
$ \lambda_2 $	1.5010	2.0001	2.5008	1.5026	2.0018	2.5009	1.5015	2.0022	2.4996
$ \sigma(\lambda_2) $	0.0019	0.0019	0.0019	0.0016	0.0016	0.0016	0.0015	0.0015	0.0015
$ \lambda_3 $	-1.4997	-2.0004	-2.4985	-1.4992	-1.9986	-2.4994	-1.5012	-2.0000	-2.5017
$ \sigma(\lambda_3) $	0.0019	0.0019	0.0019	0.0017	0.0016	0.0016	0.0015	0.0015	0.0015
$ \lambda_4 $	-0.0515	-0.0510	-0.0503	-0.1017	-0.1009	-0.1011	-0.1530	-0.1532	-0.1513
$ \sigma(\lambda_4) $	0.0019	0.0019	0.0019	0.0016	0.0016	0.0016	0.0015	0.0015	0.0015
Identified systems without group sparsity
$ \lambda_1 $	-0.0493	-0.0494	-0.0495	-0.1004	-0.1007	-0.0999	-0.1499	-0.1491	-0.1505
$ \sigma(\lambda_1) $	0.0019	0.0019	0.0019	0.0017	0.0016	0.0016	0.0015	0.0015	0.0015
$ \lambda_2 $	1.5002	1.9997	2.5007	1.5008	2.0004	2.4998	1.4991	2.0000	2.4978
$ \sigma(\lambda_2) $	0.0019	0.0019	0.0019	0.0016	0.0016	0.0016	0.0015	0.0015	0.0015
$ {C(\text{const1}))} $	0	-0.0424	-0.0524	0	0	-0.0370	0	0	0
$ \sigma({C(\text{const1}))} $	0	0.0017	0.0017	0	0	0.0013	0	0	0
$ \lambda_3 $	-1.4997	-2.0004	-2.4986	-1.4992	-1.9987	-2.4994	-1.5012	-1.9999	-2.5016
$ \sigma(\lambda_3) $	0.0019	0.0019	0.0019	0.0017	0.0016	0.0016	0.0015	0.0015	0.0015
$ \lambda_4 $	-0.0507	-0.0506	-0.0502	-0.0996	-0.0993	-0.0999	-0.1499	-0.1507	-0.1493
$ \sigma(\lambda_4) $	0.0019	0.0019	0.0019	0.0016	0.0016	0.0016	0.0015	0.0015	0.0015
$ {C(\text{const2}))} $	0	0.0440	0.0551	0	0	0.0411	0	0	0
$ \sigma({C(\text{const2}))}) $	0	0.0017	0.0017	0	0	0.0013	0	0	0
True systems
$ \lambda_1 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15
$ \lambda_2 $	1.5	2.0	2.5	1.5	2.0	2.5	1.5	2.0	2.5
$ \lambda_3 $	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5
$ \lambda_4 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15

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Table 2. Different metrics for the parametric linear systems, $ \frac{dx}{dt} = \lambda_1x + \lambda_2y $, $ \frac{dy}{dt} = \lambda_3x + \lambda_4y $. ($ 5\% $ data noise)

Identified systems with group sparsity
Case	1	2	3	4	5	6	7	8	9
$ \textbf{rmse} $	0.90	0.44	0.51	1.50	0.91	0.44	1.66	1.41	0.62
$ \mathbf{M_{R}} $	1	1	1	1	1	1	1	1	1
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1
Identified systems without group sparsity
$ \textbf{rmse} $	0.50	21.59	21.51	0.58	0.58	15.62	0.70	0.41	0.81
$ \mathbf{M_{R}} $	1	0.67	0.67	1	1	0.67	1	1	1
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1
Identified systems with non-parsimonious model
$ \textbf{rmse} $	1414.37	1417.92	1425.69	1415.33	1412.46	1417.82	1422.44	1419.52	1411.64
$ \mathbf{M_{R}} $	0.095	0.095	0.095	0.095	0.095	0.095	0.095	0.095	0.095
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1

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Table 3. Different metrics for the parametric cubic systems $ \frac{dx}{dt} = \lambda_1x^3 + \lambda_2y^3 $, $ \frac{dy}{dt} = \lambda_3x^3 + \lambda_4y^3 $ (5% data noise)

Identified systems with group sparsity
Case	1	2	3	4	5	6	7	8	9
$ \textbf{rmse} $	5.32	2.05	2.86	8.03	6.38	3.97	11.62	8.64	6.14
$ \mathbf{M_{R}} $	1	1	1	1	1	1	1	1	1
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1
Identified systems without group sparsity
$ \textbf{rmse} $	61.16	61.25	60.07	48.28	50.58	46.30	40.66	46.12	45.36
$ \mathbf{M_{R}} $	0.67	0.67	0.67	0.67	0.67	0.57	0.67	0.5	0.57
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1
Identified systems with non-parsimonious model
$ \textbf{rmse} $	64.10	72.08	70.49	56.96	51.59	55.12	50.09	52.25	46.78
$ \mathbf{M_{R}} $	0.095	0.095	0.095	0.095	0.095	0.095	0.095	0.095	0.095
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1

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Table 4. System identification results for the parametric cubic systems $ \frac{dx}{dt} = \lambda_1x^3 + \lambda_2y^3 $, state $ \frac{dy}{dt} = \lambda_3x^3 + \lambda_4y^3 $ (5% data noise), where $ \lambda_i $ denotes relevant coefficients, $ C(\square) $ denotes the coefficient of redundant terms $ \square $

Identified systems with group sparsity
Case	1	2	3	4	5	6	7	8	9
$ \lambda_1 $	-0.0458	-0.0471	-0.0468	-0.0956	-0.0951	-0.0953	-0.1456	-0.1428	-0.1448
$ \sigma(\lambda_1) $	0.0111	0.0144	0.0177	0.0073	0.0101	0.0114	0.0054	0.0085	0.0097
$ \lambda_2 $	1.5028	2.0037	2.4908	1.5093	2.0034	2.4931	1.5167	2.0100	2.5086
$ \sigma(\lambda_2) $	0.0106	0.0141	0.0177	0.0066	0.0096	0.0112	0.0047	0.0078	0.0094
$ \lambda_3 $	-1.5086	-2.0034	-2.5001	-1.5132	-2.0170	-2.5113	-1.5167	-2.0209	-2.5190
$ \sigma(\lambda_3) $	0.0106	0.0139	0.0173	0.0070	0.0104	0.0117	0.0054	0.0074	0.0092
$ \lambda_4 $	-0.0553	-0.0496	-0.0470	-0.1032	-0.1014	-0.0989	-0.1560	-0.1536	-0.1467
$ \sigma(\lambda_4) $	0.0101	0.0136	0.0172	0.0064	0.0099	0.0115	0.0046	0.0068	0.0089
Identified systems without group sparsity
$ \lambda_1 $	-0.0474	-0.0493	-0.0486	-0.0982	-0.1007	-0.1007	-0.1488	-0.1496	-0.1450
$ \sigma(\lambda_1) $	0.0021	0.0033	0.0050	0.0016	0.0027	0.0034	0.0012	0.0029	0.0031
$ \lambda_2 $	1.4991	2.0037	2.5034	1.5019	1.9898	2.4886	1.5034	2.0037	2.5127
$ \sigma(\lambda_2) $	0.0024	0.0043	0.0068	0.0020	0.0034	0.0045	0.0015	0.0034	0.0038
$ C(y^2) $	-0.0921	-0.1260	-0.1530	-0.0737	-0.0966	-0.1098	-0.0603	-0.0849	-0.1174
$ \sigma(C(y^2)) $	0.0029	0.0050	0.0078	0.0022	0.0038	0.0045	0.0017	0.0037	0.0042
$ C(x^2y) $	0	0	0	0	0	0.0405	0	0.415	0
$ \sigma(C(x^2y)) $	0	0	0	0	0	0.0086	0	0.0068	0
$ C(x^2y^2) $	0	0	0	0	0	0	0	0	0.0416
$ \sigma(C(x^2y^2)) $	0	0	0	0	0	0	0	0	0.0051
$ C(x^2y^3) $	0	0	0	0	0	0	0	-0.0381	0
$ \sigma(C(x^2y^3)) $	0	0	0	0	0	0	0	0.0068	0
$ \lambda_3 $	-1.5033	-1.9984	-2.4967	-1.5000	-2.0028	-2.4993	-1.4975	-2.0010	-2.5003
$ \sigma(\lambda_3) $	0.0026	0.0041	0.0054	0.0018	0.0028	0.0044	0.0015	0.0027	0.0030
$ \lambda_4 $	-0.0547	-0.0509	-0.0490	-0.1008	-0.1017	-0.1001	-0.1517	-0.1531	-0.1475
$ \sigma(\lambda_4) $	0.0022	0.0035	0.0046	0.0014	0.0023	0.0038	0.0011	0.0022	0.0026
$ C(x^2) $	0.0912	0.1189	0.1471	0.0714	0.1052	0.1137	0.0621	0.0819	0.1005
$ \sigma(C(x^2)) $	0.0026	0.0041	0.0054	0.0018	0.0028	0.0044	0.0015	0.0027	0.0030
True systems
$ \lambda_1 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15
$ \lambda_2 $	1.5	2.0	2.5	1.5	2.0	2.5	1.5	2.0	2.5
$ \lambda_3 $	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5
$ \lambda_4 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15

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Table 5. Different metrics for parametric Michaelisâ€"Menten model (2% data noise)

Identified systems with group sparsity
Case	1	2	3	4	5	6	7	8	9	10
$ \textbf{rmse} $	42.74	9.53	32.39	30.08	12.99	13.40	17.58	12.23	33.10	35.94
$ \mathbf{M_{R}} $	1	1	1	1	1	1	1	1	1	1
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1	1
Identified systems with non-parsimonious model
$ \textbf{rmse} $	776.67	704.91	728.43	716.23	625.28	606.92	603.25	748.48	708.18	754.67
$ \mathbf{M_{R}} $	0.43	0.43	0.43	0.43	0.43	0.43	0.43	0.43	0.43	0.43
$ \mathbf{M_{P}} $	1	1	1	1	1	1	1	1	1	1

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Table 6. Identification results for parametric Michaelisâ€"Menten model (2% data noise), where $ \lambda_i $ denotes the coefficient and $ \sigma(\lambda_i) $ denotes standard deviation for the parsimonious library terms

Identified systems with group sparsity
Term	Coeff	1	2	3	4	5	6	7	8	9	10
$ dx/dt $	$ \lambda_1 $	-0.1608	-0.1926	-0.2607	-0.3621	-0.2077	-0.2870	-0.3776	-0.1872	-0.2591	-0.3541
	$ \sigma(\lambda_1) $	0.0288	0.0203	0.226	0.0253	0.0166	0.163	0.0198	0.0199	0.0256	0.0263
$ x $	$ \lambda_2 $	-0.8985	-1.2028	-1.1931	-1.1984	-1.5070	-1.5002	-1.4989	-1.2025	-1.1962	-1.1962
	$ \sigma(\lambda_2) $	0.0097	0.0088	0.0096	0.0105	0.0087	0.0085	0.0098	0.0088	0.0107	0.0110
$ {\text{constant}} $	$ {\lambda_3} $	0.0645	0.0516	0.0851	0.1154	0.0433	0.0741	0.1043	0.1127	0.1754	0.2354
	$ \sigma({\lambda_3}) $	0.0154	0.0116	0.0121	0.0127	0.0098	0.0090	0.0105	0.0109	0.0131	0.0127
True systems
$ dx/dt $	$ \lambda_1 $	-0.2	-0.2	-0.3	-0.4	-0.2	-0.3	-0.4	-0.2	-0.3	-0.4
$ x $	$ \lambda_2 $	-0.9	-1.2	-1.2	-1.2	-1.5	-1.5	-1.5	-1.2	-1.2	-1.2
$ \text{constant} $	$ {\lambda_3} $	0.06	0.06	0.09	0.12	0.06	0.09	0.12	0.12	0.18	0.24

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Table 7. Control variables for parametric system identification

Parameter	$ C_{an} $	$ R_{f} $	$ k_{E} $
$ 1 $	$ 0.15 $	$ 2.38\times10^{3} $	$ 0.231 $
$ 2 $	$ 0.125 $	$ 2.38\times10^{3} $	$ 0.231 $
$ 3 $	$ 0.15 $	$ 2.5\times10^{3} $	$ 0.231 $
$ 4 $	$ 0.15 $	$ 2.38\times10^{3} $	$ 0.2 $
$ 5 $	$ 0.15 $	$ 2.38\times10^{3} $	$ 0.175 $

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Table 8. Right hand-side library and true parametric coefficient

True systems
Term	Coeff	1	2	3	4	5
$ P_{ic}P_aR_a\frac{d{C}_a}{dt} $	$ \lambda_1 $	0.231	0.231	0.231	0.2	0.175
$ P_{ic}^2R_a\frac{d{C}_a}{dt} $	$ \lambda_2 $	-0.231	-0.231	-0.231	-0.2	-0.175
$ P_aP_{ic} $	$ \lambda_3 $	$ 1.2035\times10^{-4} $	$ 1.2035\times10^{-4} $	$ 1.1458\times10^{-4} $	$ 1.0420\times10^{-4} $	$ 9.118\times10^{-5} $
$ P_{ic}^2 $	$ \lambda_4 $	$ -1.6571\times10^{-4} $	$ -1.6571\times10^{-4} $	$ -1.5993\times10^{-4} $	$ -1.4347\times10^{-4} $	$ -1.2554\times10^{-4} $
$ P_{ic}^2Ra $	$ \lambda_5 $	$ -3.6576\times10^{-5} $	$ -3.6576\times10^{-5} $	$ -3.6576\times10^{-5} $	$ -3.1668\times10^{-5} $	$ -2.7709\times10^{-5} $
$ P_{ic}Ra $	$ \lambda_6 $	$ 2.1946\times10^{-4} $	$ 2.1946\times10^{-4} $	$ 2.1946\times10^{-4} $	$ 1.9001\times10^{-4} $	$ 1.6625\times10^{-4} $
$ P_{ic} $	$ \lambda_7 $	$ 2.7213\times10^{-4} $	$ 2.7213\times10^{-4} $	$ 2.7213\times10^{-4} $	$ 2.3561\times10^{-4} $	$ 2.0616\times10^{-4} $
$ \frac{dP_{ic}}{dt} $	$ \lambda_8 $	$ -1.24 $	$ -1.24 $	$ -1.24 $	$ -1.24 $	$ -1.24 $
$ C_aP_{ic}R_a\frac{d{P}_{ic}}{dt} $	$ \lambda_9 $	$ -0.231 $	$ -0.231 $	$ -0.231 $	$ -0.2 $	$ -0.175 $
$ C_aP_{ic}\frac{d{P}_{ic}}{dt} $	$ \lambda_{10} $	$ -0.2864 $	$ -0.2864 $	$ -0.2864 $	$ -0.248 $	$ -0.217 $
$ P_aP_{ic}\frac{d{C_{a}}}{dt} $	$ \lambda_{11} $	$ -0.2864 $	$ -0.2864 $	$ -0.2864 $	$ -0.248 $	$ -0.217 $
$ P_{ic}^2\frac{d{C_{a}}}{dt} $	$ \lambda_{12} $	$ -0.2864 $	$ -0.2864 $	$ -0.2864 $	$ -0.248 $	$ -0.217 $

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Table 9. Different metrics for parametric ICP model

Identified systems with group sparsity
Case	1	2	3	4	5
$ \textbf{rmse} $	406.25	407.79	407.79	359.38	351.82
$ \mathbf{M_{R}} $	1	1	1	1	1
$ \mathbf{M_{P}} $	1	1	1	1	0.92
Identified systems with non-parsimonious model
$ \textbf{rmse} $	957.18	956.50	957.26	966.70	973.60
$ \mathbf{M_{R}} $	0.79	0.77	0.79	0.77	0.77
$ \mathbf{M_{P}} $	0.92	0.83	0.92	0.83	0.83

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Table 10. Right hand-side library and inferred parametric coefficient, where $ \lambda_i $ denotes the coefficient and $ \sigma(\lambda_i) $ denotes standard deviation for the parsimonious library terms

Identified systems with group sparsity
Term	Coeff	1	2	3	4	5
$ P_{ic}P_aR_a\frac{d{C}_a}{dt} $	$ \lambda_1 $	0.231	0.231	0.231	0.2	0.175
$ P_{ic}P_aR_a\frac{d{C}_a}{dt} $	$ \sigma(\lambda_1) $	$ 2.3745\times10^{-4} $	$ 5.9450\times10^{-4} $	$ 2.7723\times10^{-4} $	$ 3.4780\times10^{-4} $	$ 3.9709\times10^{-4} $
$ P_{ic}^2R_a\frac{d{C}_a}{dt} $	$ \lambda_2 $	-0.231	-0.2307	-0.231	-0.2	-0.1748
$ P_{ic}^2R_a\frac{d{C}_a}{dt} $	$ \sigma(\lambda_2) $	$ 5.9808\times10^{-4} $	$ 2.02\times10^{-2} $	$ 8.626\times10^{-4} $	$ 3\times10^{-3} $	$ 7.6\times10^{-3} $
$ P_aP_{ic} $	$ \lambda_3 $	$ 1.1972\times10^{-4} $	$ 1.1905\times10^{-4} $	$ 1.1421\times10^{-4} $	$ 1.0319\times10^{-4} $	$ 8.886\times10^{-5} $
$ P_aP_{ic} $	$ \sigma(\lambda_3) $	$ 2.62\times10^{-5} $	$ 5.881\times10^{-5} $	$ 3.841\times10^{-5} $	$ 6.221\times10^{-5} $	$ 1.6\times10^{-5} $
$ P_{ic}^2 $	$ \lambda_4 $	$ -1.6358\times10^{-4} $	$ -1.6118\times10^{-4} $	$ -1.5867\times10^{-4} $	$ -1.4143\times10^{-4} $	$ -1.1745\times10^{-4} $
$ P_{ic}^2 $	$ \sigma(\lambda_4) $	$ 6.862\times10^{-5} $	$ 1.4322\times10^{-4} $	$ 1.0414\times10^{-4} $	$ 1.3692\times10^{-4} $	$ 2.907\times10^{-5} $
$ P_{ic}^2Ra $	$ \lambda_5 $	$ -3.6628\times10^{-5} $	$ -3.6622\times10^{-5} $	$ -3.6604\times10^{-4} $	$ -3.1036\times10^{-5} $	$ -2.7871\times10^{-5} $
$ P_{ic}^2Ra $	$ \sigma(\lambda_5) $	$ 2.364\times10^{-6} $	$ 1.1038\times10^{-5} $	$ 2.988\times10^{-6} $	$ 4.857\times10^{-5} $	$ 6.661\times10^{-6} $
$ P_{ic}Ra $	$ \lambda_6 $	$ 2.2857\times10^{-4} $	$ 2.3723\times10^{-4} $	$ 2.2446\times10^{-4} $	$ 2.0615\times10^{-4} $	$ 1.9794\times10^{-4} $
$ P_{ic}Ra $	$ \sigma(\lambda_6) $	$ 4.2186\times10^{-4} $	$ 8.3757\times10^{-4} $	$ 5.9635\times10^{-4} $	$ 1.1\times10^{-3} $	$ 1.8554\times10^{-4} $
$ P_{ic} $	$ \lambda_7 $	$ 2.1870\times10^{-4} $	$ 1.5004\times10^{-4} $	$ 2.4190\times10^{-4} $	$ 1.3637\times10^{-4} $	$ 0 $
$ P_{ic} $	$ \sigma(\lambda_7) $	$ 2\times10^{-3} $	$ 4.7\times10^{-3} $	$ 3.1\times10^{-3} $	$ 5.6\times10^{-3} $	$ 0 $
$ \frac{dP_{ic}}{dt} $	$ \lambda_8 $	$ -1.2071 $	$ -1.1633 $	$ -1.2195 $	$ -1.1759 $	$ -1.0668 $
$ \frac{dP_{ic}}{dt} $	$ \sigma(\lambda_8) $	$ 0.9734 $	$ 1.8386 $	$ 1.5444 $	$ 2.4216 $	$ 0.4493 $
$ C_aP_{ic}R_a\frac{d{P}_{ic}}{dt} $	$ \lambda_9 $	$ -0.231 $	$ -0.2312 $	$ -0.231 $	$ -0.2 $	$ -0.1751 $
$ C_aP_{ic}R_a\frac{d{P}_{ic}}{dt} $	$ \sigma(\lambda_9) $	$ 5.1\times10^{-4} $	$ 1.19\times10^{-2} $	$ 7.4645\times10^{-4} $	$ 2.4\times10^{-3} $	$ 5.4\times10^{-3} $
$ C_aP_{ic}\frac{d{P_{ic}}}{dt} $	$ \lambda_{10} $	$ -0.2789 $	$ -0.2678 $	$ -0.2817 $	$ -0.2351 $	$ -0.1859 $
$ C_aP_{ic}\frac{d{P_{ic}}}{dt} $	$ \sigma(\lambda_{10}) $	$ 0.2246 $	$ 0.4533 $	$ 0.3567 $	$ 0.4868 $	$ 8.89\times10^{-2} $
$ P_aP_{ic}\frac{d{C_{a}}}{dt} $	$ \lambda_{11} $	$ -0.2789 $	$ -0.2687 $	$ -0.2817 $	$ -0.2352 $	$ -0.1867 $
$ P_aP_{ic}\frac{d{C_{a}}}{dt} $	$ \sigma(\lambda_{11}) $	$ 0.2246 $	$ 0.4246 $	$ 0.3562 $	$ 0.4843 $	$ 7.77\times10^{-2} $
$ P_{ic}^2\frac{d{C_{a}}}{dt} $	$ \lambda_{12} $	$ -0.2788 $	$ -0.2703 $	$ -0.2817 $	$ -0.2352 $	$ -0.1877 $
$ P_{ic}^2\frac{d{C_{a}}}{dt} $	$ \sigma(\lambda_{12}) $	$ 0.2249 $	$ 0.3936 $	$ 0.3562 $	$ 0.4819 $	$ 7.36\times10^{-2} $

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Table 11. Training cost comparison

	Linear	Cubic	Michaelisâ€"Menten	ICP
$ \textbf{Parsimonious} $	5s	13s	4882s	1148s
$ \textbf{Non-parsimonious} $	0.67s	0.2s	1251s	1134s

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Table 12. System identification results for the linear parametric ODE, $ \frac{dx}{dt} = \lambda_1x + \lambda_2y $, state $ \frac{dy}{dt} = \lambda_3x + \lambda_4y $ with $ 10\% $ data noise, where $ \lambda_{i} $ denotes the relevant coefficient, $ C(\text{const1}) $ denotes the redundant constant coefficient and $ \sigma(C(\text{const1})) $ denotes the standard deviation for equation $ \frac{dx}{dt} = \lambda_1x + \lambda_2y $, while $ C(\text{const2}) $ and $ \sigma(C(\text{const2})) $ are corresponding values for equation $ \frac{dy}{dt} = \lambda_3x + \lambda_4y $

Identified systems with group sparsity
Case	1	2	3	4	5	6	7	8	9
$ \lambda_1 $	-0.0506	-0.0513	-0.0522	-0.1026	-0.1042	-0.1033	-0.1517	-0.1502	-0.1539
$ \sigma(\lambda_1) $	0.0038	0.0038	0.0038	0.0033	0.0033	0.0033	0.0030	0.0030	0.0030
$ \lambda_2 $	1.5001	1.9977	2.4989	1.5034	2.0012	2.4992	1.5012	2.0017	2.4964
$ \sigma(\lambda_2) $	0.0038	0.0037	0.0037	0.0033	0.0032	0.0032	0.0029	0.0029	0.0029
$ \lambda_3 $	-1.4976	-1.9982	-2.4940	-1.4963	-1.9945	-2.4957	-1.4996	-1.9973	-2.4997
$ \sigma(\lambda_3) $	0.0038	0.0038	0.0038	0.0033	0.0033	0.0033	0.0030	0.0030	0.0030
$ \lambda_4 $	-0.0510	-0.0494	-0.0474	-0.1013	-0.0992	-0.0989	-0.1536	-0.1535	-0.1492
$ \sigma(\lambda_4) $	0.0038	0.0037	0.0037	0.0033	0.0032	0.0032	0.0029	0.0029	0.0029
Identified systems without group sparsity
$ \lambda_1 $	-0.0487	-0.0487	-0.0489	-0.1006	-0.1014	-0.1000	-0.1498	-0.1476	-0.1509
$ \sigma(\lambda_1) $	0.0038	0.0038	0.0038	0.0033	0.0033	0.0033	0.0030	0.0030	0.0030
$ \lambda_2 $	1.5005	1.9995	2.5017	1.5017	2.0008	2.5000	1.4982	1.9999	2.4958
$ \sigma(\lambda_2) $	0.0038	0.0037	0.0037	0.0033	0.0032	0.0032	0.0029	0.0029	0.0029
$ C(\text{const1}) $	-0.0626	-0.0847	-0.1048	-0.0436	-0.0608	-0.0740	0	-0.0475	-0.0597
$ \sigma(C(\text{const1})) $	0.0035	0.0035	0.0035	0.0026	0.0026	0.0026	0	0.0021	0.0021
$ \lambda_3 $	-1.4997	-2.0004	-2.4986	-1.4992	-1.9987	-2.4994	-1.5012	-1.9999	-2.5016
$ \sigma(\lambda_3) $	0.0019	0.0019	0.0019	0.0017	0.0016	0.0016	0.0015	0.0015	0.0015
$ \lambda_4 $	-0.0507	-0.0506	-0.0502	-0.0996	-0.0993	-0.0999	-0.1499	-0.1507	-0.1493
$ \sigma(\lambda_4) $	0.0019	0.0019	0.0019	0.0016	0.0016	0.0016	0.0015	0.0015	0.0015
$ C(\text{const2}) $	0	0.0440	0.0551	0	0	0.0411	0	0	0
$ \sigma(C(\text{const2})) $	0	0.0017	0.0017	0	0	0.0013	0	0	0
True systems
$ \lambda_1 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15
$ \lambda_2 $	1.5	2.0	2.5	1.5	2.0	2.5	1.5	2.0	2.5
$ \lambda_3 $	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5
$ \lambda_4 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15

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Table 13. System identification results for the parametric cubic systems $ \frac{dx}{dt} = \lambda_1x^3 + \lambda_2y^3 $, state $ \frac{dy}{dt} = \lambda_3x^3 + \lambda_4y^3 $ with 10% data noise.

Identified systems with group sparsity
Case	1	2	3	4	5	6	7	8	9
$ y^2 $	-0.1749	-0.2424	-0.2813	-0.1295	-0.1940	-0.2175	-0.1045	-0.1744	-0.2075
$ \sigma(y^2) $	0.0044	0.0061	0.0088	0.0027	0.0041	0.0050	0.0026	0.0035	0.0051
$ \lambda_1 $	-0.0485	-0.0458	-0.0503	-0.1007	-0.0978	-0.1025	-0.1498	-0.1536	-0.1525
$ \sigma(\lambda_1) $	0.0039	0.0053	0.0075	0.0025	0.0038	0.0045	0.0025	0.0033	0.0047
$ \lambda_2 $	1.5013	1.9995	2.4978	1.4994	2.0097	2.4998	1.4909	2.0109	2.5204
$ \sigma(\lambda_2) $	0.0037	0.0052	0.0075	0.0023	0.0036	0.0044	0.0022	0.0031	0.0045
$ x^2 $	0.1797	0.2177	0.3016	0.1434	0.1922	0.2480	0.1273	0.1710	0.2083
$ \sigma(x^2) $	0.0047	0.0060	0.0084	0.0032	0.0049	0.0061	0.0030	0.0037	0.0060
$ \lambda_3 $	-1.5033	-2.0000	-2.5049	-1.5059	-1.9995	-2.5155	-1.5117	-2.0026	-2.5049
$ \sigma(\lambda_3) $	0.0041	0.0051	0.0072	0.0029	0.0043	0.0054	0.0028	0.0034	0.0055
$ \lambda_4 $	-0.0508	-0.0541	-0.0514	-0.0998	-0.1056	-0.0950	-0.1473	-0.1528	-0.1485
$ \sigma(\lambda_4) $	0.0039	0.0050	0.0072	0.0026	0.0041	0.0053	0.0023	0.0030	0.0053
Identified systems without group sparsity
$ C(y^2) $	$ -0.1772 $	$ -0.2423 $	$ -0.2791 $	$ -0.1377 $	$ -0.2122 $	$ -0.2376 $	$ -0.1153 $	$ -0.1898 $	$ -0.1975 $
$ \sigma(C(y^2)) $	$ 0.0056 $	$ 0.0055 $	$ 0.0082 $	$ 0.0034 $	$ 0.0042 $	$ 0.0056 $	$ 0.0030 $	$ 0.0039 $	$ 0.0062 $
$ \lambda_1 $	-0.0489	-0.0468	-0.0514	-0.0990	-0.0931	-0.1005	-0.1476	-0.1493	-0.1548
$ \sigma(\lambda_1) $	0.0037	0.0047	0.0055	0.0024	0.0030	0.0038	0.0022	0.0030	0.0045
$ C(x^2y) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ -0.0558 $
$ \sigma(C(x^2y)) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0189 $
$ \lambda_2 $	1.5068	1.9985	2.5174	1.4950	2.0084	2.5035	1.4918	2.0150	2.5325
$ \sigma(\lambda_2) $	0.0051	0.0069	0.0083	0.0034	0.0041	0.0050	0.0022	0.0039	0.0062
$ C(x^2y^2) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0361 $	$ 0.0378 $	$ 0 $	$ 0.0365 $	$ 0 $
$ \sigma(C(x^2y^2)) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0049 $	$ 0.0062 $	$ 0 $	$ 0.0051 $	$ 0 $
$ C(x^4y) $	$ 0 $	$ 0 $	$ 0.0513 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ \sigma(C(x^4y)) $	$ 0 $	$ 0 $	$ 0.0137 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $
$ C(x^2y^3) $	0	0	-0.0589	0	0	0	0	0	0
$ \sigma(C(x^2y^3)) $	0	0	0.0125	0	0	0	0	0	0
$ C(x^2) $	$ 0.1798 $	$ 0.2060 $	$ 0.2977 $	$ 0.1418 $	$ 0.1946 $	$ 0.2289 $	$ 0.1264 $	$ 0.1610 $	$ 0.2011 $
$ \sigma(C(x^2)) $	$ 0.0050 $	$ 0.0072 $	$ 0.0092 $	$ 0.0039 $	$ 0.0042 $	$ 0.0058 $	$ 0.0023 $	$ 0.0032 $	$ 0.0059 $
$ \lambda_3 $	-1.4942	-1.9919	-2.4866	-1.5093	-2.0194	-2.5208	-1.5049	-2.0173	-2.5176
$ \sigma(\lambda_3) $	0.0050	0.0063	0.0084	0.0037	0.0039	0.0052	0.0022	0.0031	0.0056
$ C(xy^2) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ -0.0386 $	$ 0 $
$ \sigma(C(xy^2)) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0057 $	$ 0 $
$ \lambda_4 $	-0.0511	-0.0548	-0.0502	-0.0994	-0.1089	-0.0977	-0.1472	-0.1549	-0.1487
$ \sigma(\lambda_4) $	0.0035	0.0045	0.0062	0.0025	0.0027	0.0039	0.0016	0.0020	0.0042
$ C(x^3y) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0482 $	$ 0 $	$ 0 $	$ -0.0603 $
$ \sigma(C(x^3y)) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0064 $	$ 0 $	$ 0 $	$ 0.0068 $
$ C(x^2y^2) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0368 $	$ 0 $
$ \sigma(C(x^2y^2)) $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0.0044 $	$ 0 $
$ C(xy^3) $	0	0	0	0	0	0	0	0	0.0420
$ \sigma(C(xy^3)) $	0	0	0	0	0	0	0	0	0.0071
$ C(x^3y^2) $	0	0	0	0	0.0501	0	0	0.0637	0
$ \sigma(C(x^3y^2)) $	0	0	0	0	0.0073	0	0	0.0065	0
True systems
$ \lambda_1 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15
$ \lambda_2 $	1.5	2.0	2.5	1.5	2.0	2.5	1.5	2.0	2.5
$ \lambda_3 $	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5	-1.5	-2.0	-2.5
$ \lambda_4 $	-0.05	-0.05	-0.05	-0.1	-0.1	-0.1	-0.15	-0.15	-0.15

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