Comparison of Artificial Neural Network Architecture in Solving Ordinary Differential Equations
Table 4
Analytical and neural solutions with arbitrary- and regression-based weights (Example 3).
Input data
Analytical
Euler
Runge-Kutta
Neural results
(four nodes)
(four nodes)
(five nodes)
(five nodes)
(six nodes)
(six nodes)
0
0
0
0
0
0
0
0
0
0
0.1
0.0671
0.1000
0.0671
0.0440
0.0539
0.0701
0.0602
0.0565
0.0670
0.2
0.0905
0.1241
0.0904
0.0867
0.0938
0.0877
0.0927
0.0921
0.0907
0.3
0.0917
0.1169
0.0917
0.0849
0.0926
0.0889
0.0932
0.0931
0.0918
0.4
0.0829
0.0991
0.0829
0.0830
0.0876
0.0806
0.0811
0.0846
0.0824
0.5
0.0705
0.0797
0.0705
0.0760
0.0748
0.0728
0.0714
0.0717
0.0706
0.6
0.0578
0.0622
0.0577
0.0492
0.0599
0.0529
0.0593
0.0536
0.0597
0.7
0.0461
0.0476
0.0461
0.0433
0.0479
0.0410
0.0453
0.0450
0.0468
0.8
0.0362
0.0360
0.0362
0.0337
0.0319
0.0372
0.0370
0.0343
0.0355
0.9
0.0280
0.0271
0.0280
0.0324
0.0308
0.0309
0.0264
0.0249
0.0284
1.0
0.0215
0.0203
0.0215
0.0304
0.0282
0.0255
0.0247
0.0232
0.0217
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