Bio-inspired computational heuristics to study Lane–Emden systems arising in astrophysics model

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Table 2 Best weights trained for neural network modelling by SQP, PS and PSO-SQP algorithms

Methods	I	\(\delta_{i}\)	β _i	_{ω i}	i	δ i _i	β _i	_{ω i}
SQP	1	0.78636	0.11707	−1.72866	6	2.24414	−1.07067	3.55435
	2	−2.80683	0.84866	−1.47352	7	0.72638	0.80491	−1.05825
	3	−0.66561	0.11462	−1.56349	8	−0.35824	−0.30248	−0.81978
	4	−2.77591	−0.10056	−0.56814	9	−0.93043	−0.67351	0.00426
	5	0.35878	−1.54018	−0.39116	10	0.77053	1.44174	1.15517
PS	1	3.03789	−0.35385	0.02289	6	−1.38575	4.00223	−8.80831
	2	0.82521	−0.82359	1.73409	7	1.10965	−0.74481	−1.14468
	3	−8.99993	−3.18106	−4.91916	8	−0.27787	−3.01028	0.61488
	4	−1.05818	0.54020	1.74266	9	−5.93029	0.35017	−2.00264
	5	7.54212	0.28198	−0.83137	10	−2.05182	0.45484	2.87243
PS-SQP	1	−0.72218	0.55533	3.06589	6	−1.26715	1.30549	−1.92557
	2	0.70591	3.76635	4.43837	7	−1.18046	−1.15109	−0.18455
	3	0.09391	3.18733	1.15390	8	0.00790	1.08650	0.35001
	4	7.63559	−0.07725	−1.55487	9	0.06668	−4.96006	7.78676
	5	1.38422	−0.76144	−1.69561	10	0.00121	−6.38647	6.67553
GA	1	1.29680	−0.70586	0.94321	6	0.18766	2.70748	−1.62678
	2	−1.08867	0.43791	1.16502	7	0.31812	2.33138	−0.61683
	3	0.95837	−0.89643	1.47983	8	−0.06982	0.73937	−0.09366
	4	0.415759	3.43301	1.48746	9	−0.55372	0.98114	1.17121
	5	−0.01698	5.29995	−2.90769	10	0.14876	1.78333	0.46163
GA-SQP	1	−0.17494	1.78344	0.98665	6	−0.19778	0.02527	0.14497
	2	−0.91882	−0.05487	0.84314	7	−0.18451	−0.15034	0.50728
	3	−0.07822	0.34068	0.50753	8	0.32558	−1.24882	2.01797
	4	0.75215	1.77652	1.46356	9	2.07449	−1.21908	3.30059
	5	−0.08314	1.50680	−1.73595	10	−1.08201	0.038511	1.28033