Using Feed Forward Neural Network to Solve Eigenvalue Problems

Table 1

Analytic and neural solution of Example 1.

Input	Analytic solution	Output of suggested FFNN 𝑦𝑡(𝑥) for different training algorithms
𝑥	𝑦𝑎(𝑥)	Trainlm	Trainbfg	Trainbr
0.0	1	1.000000000000000	1.000000000347198	1.000000000000027
0.1	0.223130160148430	0.223130160148430	0.223130160475769	0.223130160148043
0.2	0.002478752176666	0.002478752176666	0.002478752608702	0.002478752184984
0.3	1.370959086384080𝑒−06	1.370959086255397𝑒−06	2.138822309394328𝑒−05	1.370752687690491𝑒−06
0.4	3.775134544279084𝑒−11	−1.423704123482139𝑒−07	9.137549916715670𝑒−08	2.862885595256159𝑒−09
0.5	5.175555005801869𝑒−17	2.220446049250313𝑒−16	−4.773932360535582𝑒−09	−1.930348370038360𝑒−08
0.6	3.532628572200807𝑒−24	4.440892098500626𝑒−16	1.076577582637128𝑒−08	6.462875423718373𝑒−08
0.7	1.200481799513899𝑒−32	−2.716475933084439𝑒−10	−6.335612035002214𝑒−09	−9.347123508529620𝑒−08
0.8	2.031092662734782𝑒−42	−1.559072870804812𝑒−10	−2.091692152816904𝑒−08	2.220824357745954𝑒−08
0.9	1.710883542651365𝑒−53	7.175095973164410𝑒−66	6.814833142243515𝑒−10	6.371296168428131𝑒−08
1.0	7.175095973164410𝑒−66	1.589879339292111𝑒−10	8.929697603576870𝑒−08	−4.040193102294865𝑒−08