Artificial neural network study of the electrical conductivity of mould flux

Figures & data

Figure 1. Schematic diagram illustrates 2-layer neural network.

Table 1. The subsystems in the electrical conductivity database.

No. Components	System	Reference
1	CaF2	[3,6]
2	CaO + SiO2	[8,23]
2	CaO + Al2O3	[4,8]
2	CaO + CaF2	[3,4,8]
2	Al2O3 + CaF2	[3,4]
2	MgO + CaF2	[4]
3	CaO + SiO2 + Al2O3	[5,7]
3	CaO + SiO2 + NaO	[23,24]
3	CaO + SiO2 + K2O	[23]
3	CaO + SiO2 + CaF2	[6]
3	CaO + Al2O3 + CaF2	[3]
3	SiO2 + Al2O3 + NaO	[24]
3	SiO2 + NaO + FeO	[24]
3	SiO2 + NaO + MgO	[24]
3	SiO2 + NaO + MnO	[24]
4	CaO + SiO2 + Al2O3 + NaO	[24]
4	CaO + SiO2 + NaO + K2O	[23]
4	CaO + SiO2 + Al2O3 + K2O	[23]
4	CaO + SiO2 + Al2O3 + MgO	[23]
5	CaO + SiO2 + Al2O3 + NaO + K2O	[4,5,7]
5	CaO + SiO2 + Al2O3 + CaF2 + FeO	[23]
6	CaO + SiO2 + Al2O3 + MgO + CaF2 + FeO	[22]
6	CaO + SiO2 + Al2O3 + MgO + FeO + MnO	[4,22]
8	CaO + SiO2 + Al2O3 + NaO + K2O + MgO + CaF2 + Cr2O3	[21]

Table 2. The range of parameters’ values in the database.

					(at.-%)						(K)
Value	CaO	SiO2	Al2O3	NaO	K2O	MgO	CaF2	Cr2O3	FeO	MnO	T
Min	0	0	0	0	0	0	0	0	0	0	471
Max	72.24	60.81	43.24	29.21	6.00	20.00	100.00	0.523	34.80	20.00	2039

Figure 2. The evolution of total difference ($E_D$), regularisation term ($E_W$) and target function ($E$) vs. iteration steps for 2-layer neural network with hidden layer containing 16 units.

Figure 3. The evolution of total difference ($E_D$), regulation term ($E_W$) and target function ($E$) vs. number of units in hidden layer after 1.6 × 10⁷ iteration steps.

Table 3. The optimised weight factors and biases obtained by artificial neural network calculations.

$\mathit{i}$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
${\mathit{\omega }}_{1,\mathit{i}}^{(1)}$	−3.522	−1.679	4.007	−1.131	−4.703	1.314	−3.504	−1.977	−1.589	−2.611	−3.141
${\mathit{\omega }}_{2,\mathit{i}}^{(1)}$	2.222	−0.300	−1.014	2.707	5.379	−4.533	3.134	5.043	−6.042	1.987	6.161
${\mathit{\omega }}_{3,\mathit{i}}^{(1)}$	2.046	−4.629	−0.695	−6.015	3.567	−0.802	0.999	9.960	−0.642	−4.569	0.939
${\mathit{\omega }}_{4,\mathit{i}}^{(1)}$	0.932	2.293	0.716	4.157	0.552	−3.415	−1.466	−6.693	−3.632	0.258	−6.973
${\mathit{\omega }}_{5,\mathit{i}}^{(1)}$	−2.666	10.377	−0.491	−0.494	−4.455	−5.990	1.666	−1.483	−4.617	−2.839	−0.185
${\mathit{\omega }}_{6,\mathit{i}}^{(1)}$	2.605	−1.698	−2.027	−3.533	3.922	3.881	−5.338	0.311	−0.906	−4.354	−0.171
${\mathit{\omega }}_{7,\mathit{i}}^{(1)}$	4.777	2.718	−4.250	0.815	7.090	5.123	−3.763	6.540	−4.007	−0.648	3.776
${\mathit{\omega }}_{8,\mathit{i}}^{(1)}$	−7.403	2.192	1.479	3.087	−5.408	0.885	−1.703	−6.276	3.248	−0.600	−6.350
${\mathit{\omega }}_{9,\mathit{i}}^{(1)}$	−3.031	−0.087	5.793	−0.693	−1.810	−3.520	5.388	−0.979	−3.414	1.390	−3.816
${\mathit{\omega }}_{10,\mathit{i}}^{(1)}$	−5.049	3.333	−2.414	−3.978	0.912	1.814	4.518	0.182	2.548	2.945	−1.225
${\mathit{\omega }}_{11,\mathit{i}}^{(1)}$	−0.267	−1.132	−3.102	3.209	−0.603	−7.334	−3.738	0.064	5.103	2.639	5.318
${\mathit{\omega }}_{12,\mathit{i}}^{(1)}$	0.376	−2.656	−0.931	1.841	−0.312	3.175	−0.627	−6.724	−1.493	−0.382	−2.201
${\mathit{\omega }}_{13,\mathit{i}}^{(1)}$	1.860	−8.368	−5.901	6.286	−1.250	0.786	2.651	0.438	2.259	0.032	6.445
${\mathit{\omega }}_{14,\mathit{i}}^{(1)}$	−2.727	5.957	3.358	2.482	2.415	7.337	−1.195	−8.983	2.609	−0.273	−3.009
${\mathit{\omega }}_{15,\mathit{i}}^{(1)}$	−0.862	1.249	−5.116	5.106	−2.449	0.498	−3.733	−3.322	4.180	0.328	−1.869
${\mathit{\omega }}_{16,\mathit{i}}^{(1)}$	1.730	0.858	−0.612	−2.560	−0.201	−1.836	1.620	−6.329	−2.162	−1.012	1.505
${\mathit{\omega }}_{1,\mathit{i}}^{(2)}$	−0.173	9.125	0.146	3.450	−8.575	2.630	6.701	4.235	−0.306	1.920	−4.530	−10.039	8.650	−9.208	−1.216	1.997
${\mathit{b}}_{{\mathit{i}}^{(1)}}$	1.172	0.645	0.321	−2.985	4.332	−5.004	2.231	6.241	6.158	4.916	−2.561	3.300	0.997	5.489	−3.383	3.552
${\mathit{b}}_{{\mathit{i}}^{(2)}}$	−6.009

Figure 4. The electrical conductivity from numerical results by artificial neural network calculation vs. the value in the database: (a) in linear scale plotting, and (b) in logarithmic scale plotting.

Figure 5. Validation of neural network calculation by experimental data reported in literature [6,26,27]. (a) CaF₂-Al₂O₃ at 1773K, (b) CaF₂-Al₂O₃ at 1973K, (c) Cao-CaF2 at 1773K, and (d) Cao-CaF2 at 1873K.

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