用神经网络分类无理数和超越数2**0.5,3**0.5,e
制作两个神经网络用来分类2**0.5、3**0.5，2**0.5和e。每个无理数取3万位有效数字，每10个数字变成一张图片。用前2500张图片来训练网络，用2500-3000张图片来做测试。
比如2**0.5的第一张图片
收敛标准δ取0.5到1e-4共16个值，每个收敛标准收敛199次，统计平均分辨准确率，迭代次数。
得到的表格
|
训练集 |
0-2500 |
||||||||
|
测试集 |
2500-3000 |
||||||||
|
f2[0] |
f2[1] |
迭代次数n |
平均准确率p-ave |
δ |
耗时ms/次 |
耗时ms/199次 |
耗时 min/199 |
最大值p-max |
pave标准差 |
|
0.501147535 |
0.499689475 |
17.51758794 |
0.499537688 |
0.5 |
4.884422111 |
972 |
0.0162 |
0.522 |
0.004812811 |
|
0.422210311 |
0.577346693 |
11189.55779 |
0.483246231 |
0.4 |
104.9095477 |
20877 |
0.34795 |
0.524 |
0.013828623 |
|
0.296495826 |
0.703493738 |
34580.47236 |
0.498226131 |
0.3 |
316.3115578 |
62946 |
1.0491 |
0.506 |
0.004294922 |
|
0.266027637 |
0.733965365 |
116589.8241 |
0.486874372 |
0.2 |
1056.276382 |
210215 |
3.503583333 |
0.525 |
0.01838175 |
|
0.441127684 |
0.558872658 |
178663.6633 |
0.493236181 |
0.1 |
1204.477387 |
239691 |
3.99485 |
0.537 |
0.019346601 |
|
0.527119391 |
0.472880614 |
299744.0854 |
0.491236181 |
0.01 |
2715.798995 |
540460 |
9.007666667 |
0.53 |
0.013548702 |
|
0.57774068 |
0.422259283 |
427566.1055 |
0.49361809 |
0.001 |
3880.844221 |
772291 |
12.87151667 |
0.541 |
0.015254454 |
|
0.507523048 |
0.492476938 |
423875.9045 |
0.492979899 |
9.00E-04 |
3867.422111 |
769633 |
12.82721667 |
0.534 |
0.014838142 |
|
0.567738564 |
0.432261394 |
452748.3869 |
0.491341709 |
8.00E-04 |
4193.211055 |
834466 |
13.90776667 |
0.531 |
0.015622231 |
|
0.482439484 |
0.517560534 |
452554.5729 |
0.492432161 |
7.00E-04 |
3811.422111 |
758474 |
12.64123333 |
0.537 |
0.015850881 |
|
0.537647222 |
0.462352754 |
459869.4724 |
0.492135678 |
6.00E-04 |
4248.542714 |
845465 |
14.09108333 |
0.54 |
0.015820815 |
|
0.517569942 |
0.482430045 |
466761.0905 |
0.494291457 |
5.00E-04 |
4226.030151 |
840989 |
14.01648333 |
0.539 |
0.01634968 |
|
0.522594814 |
0.477405195 |
489836.603 |
0.491351759 |
4.00E-04 |
4419.703518 |
879525 |
14.65875 |
0.527 |
0.015002743 |
|
0.542689058 |
0.457310917 |
528504.8643 |
0.491201005 |
3.00E-04 |
4827.341709 |
960648 |
16.0108 |
0.536 |
0.013936316 |
|
0.572835882 |
0.427164091 |
569002.3518 |
0.492668342 |
2.00E-04 |
5219.713568 |
1038733 |
17.31221667 |
0.533 |
0.013133277 |
|
0.552753899 |
0.447246105 |
660039.6482 |
0.493246231 |
1.00E-04 |
6018.592965 |
1197716 |
19.96193333 |
0.53 |
0.013978615 |
|
2**0.5 |
|
|
2500-3000 |
|
|
3**0.5 |
e |
|
平均准确率p-ave |
平均准确率p-ave |
|
0.500452 |
0.499538 |
|
0.517995 |
0.483246 |
|
0.519864 |
0.498226 |
|
0.513638 |
0.486874 |
|
0.512447 |
0.493236 |
|
0.504769 |
0.491236 |
|
0.506643 |
0.493618 |
|
0.504467 |
0.49298 |
|
0.50492 |
0.491342 |
|
0.504734 |
0.492432 |
|
0.506146 |
0.492136 |
|
0.506638 |
0.494291 |
|
0.50705 |
0.491352 |
|
0.504935 |
0.491201 |
|
0.50692 |
0.492668 |
|
0.504578 |
0.493246 |
可以看到3**0.5的分类准确率显著的大于50%，而e的pave数据显著的小于50%。差异非常明显。这表明无理数的数字分布是有规律的，给出一个10位的数字序列可能存在一种方法判断这个序列属于哪个无理数。
无理数的数据来源
https://www.wolframalpha.com/input/?i=x%5E2-1
N[sqr(2),30000]