深度学习习题与总结

◇ 2.5 自动微分

▷ 5. 使f(x)=sin(x)f(x)=sin(x)，绘制f(x)f(x)和df(x)dx\frac{df(x)}{dx}的图像，其中后者不使用f′(x)=cos⁡(x)f'(x)=\cos(x)。

代码：

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import torch
import matplotlib.pyplot as plt
# import numpy

x = torch.arange(0, 10, 0.1)
x.requires_grad_(True)

y = torch.sin(x)

y.sum().backward()

print(f"x = \n {x}")
print(f"x.detach() = \n {x.detach()}")
print(f"x.detach().numpy() = \n {x.detach().numpy()}")
# print(f"x_numpy_16 = {x.detach().numpy().astype(numpy.float16)}")

print('\n','\n')

print(f"y = \n {y}")
print(f"y.detach() = \n {y.detach()}")
print(f"y.detach().numpy() = \n {y.detach().numpy()}")

print('\n','\n')

print(f"x.grad = \n {x.grad}")
print(f"x.grad.numpy() = \n {x.grad.numpy()}")

plt.plot(x.detach().numpy(), y.detach().numpy(), label='sin(x)', color='blue')
plt.plot(x.detach().numpy(), x.grad.numpy(), label='cos(x)', color='red')

plt.xlabel('x')
plt.ylabel('y')
plt.legend()    # 为图表添加图例
plt.show()

结果：

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x = 
 tensor([0.0000, 0.1000, 0.2000, 0.3000, 0.4000, 0.5000, 0.6000, 0.7000, 0.8000,
        0.9000, 1.0000, 1.1000, 1.2000, 1.3000, 1.4000, 1.5000, 1.6000, 1.7000,
        1.8000, 1.9000, 2.0000, 2.1000, 2.2000, 2.3000, 2.4000, 2.5000, 2.6000,
        2.7000, 2.8000, 2.9000, 3.0000, 3.1000, 3.2000, 3.3000, 3.4000, 3.5000,
        3.6000, 3.7000, 3.8000, 3.9000, 4.0000, 4.1000, 4.2000, 4.3000, 4.4000,
        4.5000, 4.6000, 4.7000, 4.8000, 4.9000, 5.0000, 5.1000, 5.2000, 5.3000,
        5.4000, 5.5000, 5.6000, 5.7000, 5.8000, 5.9000, 6.0000, 6.1000, 6.2000,
        6.3000, 6.4000, 6.5000, 6.6000, 6.7000, 6.8000, 6.9000, 7.0000, 7.1000,
        7.2000, 7.3000, 7.4000, 7.5000, 7.6000, 7.7000, 7.8000, 7.9000, 8.0000,
        8.1000, 8.2000, 8.3000, 8.4000, 8.5000, 8.6000, 8.7000, 8.8000, 8.9000,
        9.0000, 9.1000, 9.2000, 9.3000, 9.4000, 9.5000, 9.6000, 9.7000, 9.8000,
        9.9000], requires_grad=True)
x.detach() = 
 tensor([0.0000, 0.1000, 0.2000, 0.3000, 0.4000, 0.5000, 0.6000, 0.7000, 0.8000,
        0.9000, 1.0000, 1.1000, 1.2000, 1.3000, 1.4000, 1.5000, 1.6000, 1.7000,
        1.8000, 1.9000, 2.0000, 2.1000, 2.2000, 2.3000, 2.4000, 2.5000, 2.6000,
        2.7000, 2.8000, 2.9000, 3.0000, 3.1000, 3.2000, 3.3000, 3.4000, 3.5000,
        3.6000, 3.7000, 3.8000, 3.9000, 4.0000, 4.1000, 4.2000, 4.3000, 4.4000,
        4.5000, 4.6000, 4.7000, 4.8000, 4.9000, 5.0000, 5.1000, 5.2000, 5.3000,
        5.4000, 5.5000, 5.6000, 5.7000, 5.8000, 5.9000, 6.0000, 6.1000, 6.2000,
        6.3000, 6.4000, 6.5000, 6.6000, 6.7000, 6.8000, 6.9000, 7.0000, 7.1000,
        7.2000, 7.3000, 7.4000, 7.5000, 7.6000, 7.7000, 7.8000, 7.9000, 8.0000,
        8.1000, 8.2000, 8.3000, 8.4000, 8.5000, 8.6000, 8.7000, 8.8000, 8.9000,
        9.0000, 9.1000, 9.2000, 9.3000, 9.4000, 9.5000, 9.6000, 9.7000, 9.8000,
        9.9000])
x.detach().numpy() = 
 [0.         0.1        0.2        0.3        0.4        0.5
 0.6        0.7        0.8        0.90000004 1.         1.1
 1.2        1.3        1.4        1.5        1.6        1.7
 1.8000001  1.9        2.         2.1        2.2        2.3
 2.4        2.5        2.6000001  2.7        2.8000002  2.9
 3.         3.1000001  3.2        3.3        3.4        3.5
 3.6000001  3.7        3.8        3.9        4.         4.1
 4.2        4.3        4.4        4.5        4.6        4.7
 4.8        4.9        5.         5.1000004  5.2000003  5.3
 5.4        5.5        5.6        5.7        5.7999997  5.9
 6.         6.1        6.2        6.2999997  6.4        6.5
 6.6        6.7000003  6.8        6.9        7.         7.1
 7.2        7.2999997  7.3999996  7.5        7.6        7.7
 7.7999997  7.8999996  8.         8.1        8.2        8.3
 8.4        8.5        8.6        8.7        8.8        8.900001
 9.         9.1        9.2        9.3        9.400001   9.5
 9.6        9.7        9.8        9.9       ]

 

y = 
 tensor([ 0.0000,  0.0998,  0.1987,  0.2955,  0.3894,  0.4794,  0.5646,  0.6442,
         0.7174,  0.7833,  0.8415,  0.8912,  0.9320,  0.9636,  0.9854,  0.9975,
         0.9996,  0.9917,  0.9738,  0.9463,  0.9093,  0.8632,  0.8085,  0.7457,
         0.6755,  0.5985,  0.5155,  0.4274,  0.3350,  0.2392,  0.1411,  0.0416,
        -0.0584, -0.1577, -0.2555, -0.3508, -0.4425, -0.5298, -0.6119, -0.6878,
        -0.7568, -0.8183, -0.8716, -0.9162, -0.9516, -0.9775, -0.9937, -0.9999,
        -0.9962, -0.9825, -0.9589, -0.9258, -0.8835, -0.8323, -0.7728, -0.7055,
        -0.6313, -0.5507, -0.4646, -0.3739, -0.2794, -0.1822, -0.0831,  0.0168,
         0.1165,  0.2151,  0.3115,  0.4049,  0.4941,  0.5784,  0.6570,  0.7290,
         0.7937,  0.8504,  0.8987,  0.9380,  0.9679,  0.9882,  0.9985,  0.9989,
         0.9894,  0.9699,  0.9407,  0.9022,  0.8546,  0.7985,  0.7344,  0.6630,
         0.5849,  0.5010,  0.4121,  0.3191,  0.2229,  0.1245,  0.0248, -0.0752,
        -0.1743, -0.2718, -0.3665, -0.4575], grad_fn=<SinBackward0>)
y.detach() = 
 tensor([ 0.0000,  0.0998,  0.1987,  0.2955,  0.3894,  0.4794,  0.5646,  0.6442,
         0.7174,  0.7833,  0.8415,  0.8912,  0.9320,  0.9636,  0.9854,  0.9975,
         0.9996,  0.9917,  0.9738,  0.9463,  0.9093,  0.8632,  0.8085,  0.7457,
         0.6755,  0.5985,  0.5155,  0.4274,  0.3350,  0.2392,  0.1411,  0.0416,
        -0.0584, -0.1577, -0.2555, -0.3508, -0.4425, -0.5298, -0.6119, -0.6878,
        -0.7568, -0.8183, -0.8716, -0.9162, -0.9516, -0.9775, -0.9937, -0.9999,
        -0.9962, -0.9825, -0.9589, -0.9258, -0.8835, -0.8323, -0.7728, -0.7055,
        -0.6313, -0.5507, -0.4646, -0.3739, -0.2794, -0.1822, -0.0831,  0.0168,
         0.1165,  0.2151,  0.3115,  0.4049,  0.4941,  0.5784,  0.6570,  0.7290,
         0.7937,  0.8504,  0.8987,  0.9380,  0.9679,  0.9882,  0.9985,  0.9989,
         0.9894,  0.9699,  0.9407,  0.9022,  0.8546,  0.7985,  0.7344,  0.6630,
         0.5849,  0.5010,  0.4121,  0.3191,  0.2229,  0.1245,  0.0248, -0.0752,
        -0.1743, -0.2718, -0.3665, -0.4575])
y.detach().numpy() = 
 [ 0.          0.09983342  0.19866933  0.29552022  0.38941836  0.47942555
  0.5646425   0.64421767  0.7173561   0.7833269   0.84147096  0.8912074
  0.9320391   0.9635582   0.98544973  0.997495    0.9995736   0.9916648
  0.9738476   0.9463001   0.9092974   0.8632094   0.80849636  0.74570525
  0.6754631   0.5984721   0.51550126  0.42737985  0.33498797  0.23924924
  0.14112     0.04158052 -0.05837419 -0.15774564 -0.2555412  -0.35078323
 -0.4425206  -0.5298362  -0.61185783 -0.68776625 -0.7568025  -0.81827706
 -0.87157565 -0.916166   -0.9516021  -0.9775301  -0.99369097 -0.9999232
 -0.99616456 -0.9824526  -0.9589243  -0.92581457 -0.8834545  -0.83226734
 -0.77276444 -0.7055403  -0.6312667  -0.5506857  -0.46460244 -0.37387657
 -0.2794155  -0.1821626  -0.08308959  0.01681362  0.1165493   0.21511999
  0.3115413   0.40485018  0.4941135   0.57843983  0.6569866   0.728969
  0.79366773  0.85043645  0.8987079   0.93799996  0.96791965  0.9881682
  0.9985433   0.99894136  0.98935825  0.9698897   0.94073063  0.90217173
  0.8545991   0.7984871   0.7343968   0.66296935  0.58491707  0.5010204
  0.4121185   0.319098    0.2228901   0.12445424  0.02477485 -0.07515112
 -0.17432715 -0.27176043 -0.3664793  -0.45753556]

 

x.grad = 
 tensor([ 1.0000,  0.9950,  0.9801,  0.9553,  0.9211,  0.8776,  0.8253,  0.7648,
         0.6967,  0.6216,  0.5403,  0.4536,  0.3624,  0.2675,  0.1700,  0.0707,
        -0.0292, -0.1288, -0.2272, -0.3233, -0.4161, -0.5048, -0.5885, -0.6663,
        -0.7374, -0.8011, -0.8569, -0.9041, -0.9422, -0.9710, -0.9900, -0.9991,
        -0.9983, -0.9875, -0.9668, -0.9365, -0.8968, -0.8481, -0.7910, -0.7259,
        -0.6536, -0.5748, -0.4903, -0.4008, -0.3073, -0.2108, -0.1122, -0.0124,
         0.0875,  0.1865,  0.2837,  0.3780,  0.4685,  0.5544,  0.6347,  0.7087,
         0.7756,  0.8347,  0.8855,  0.9275,  0.9602,  0.9833,  0.9965,  0.9999,
         0.9932,  0.9766,  0.9502,  0.9144,  0.8694,  0.8157,  0.7539,  0.6845,
         0.6084,  0.5261,  0.4385,  0.3466,  0.2513,  0.1534,  0.0540, -0.0460,
        -0.1455, -0.2435, -0.3392, -0.4314, -0.5193, -0.6020, -0.6787, -0.7486,
        -0.8111, -0.8654, -0.9111, -0.9477, -0.9748, -0.9922, -0.9997, -0.9972,
        -0.9847, -0.9624, -0.9304, -0.8892])
x.grad.numpy() = 
 [ 1.          0.9950042   0.9800666   0.9553365   0.921061    0.87758255
  0.8253356   0.7648422   0.6967067   0.6216099   0.54030234  0.45359612
  0.3623577   0.26749888  0.16996716  0.0707372  -0.02919955 -0.12884454
 -0.22720216 -0.32328954 -0.41614684 -0.50484604 -0.58850116 -0.666276
 -0.7373938  -0.8011436  -0.85688883 -0.90407217 -0.9422224  -0.9709582
 -0.9899925  -0.99913514 -0.9982948  -0.9874798  -0.9667982  -0.9364567
 -0.8967584  -0.8481     -0.79096776 -0.72593224 -0.6536436  -0.57482404
 -0.490261   -0.400799   -0.30733278 -0.2107958  -0.11215262 -0.01238885
  0.08749917  0.18651247  0.2836622   0.3779781   0.46851692  0.5543745
  0.63469297  0.7086698   0.7755658   0.8347127   0.8855194   0.9274785
  0.96017027  0.98326844  0.9965421   0.9998586   0.9931849   0.97658765
  0.9502326   0.91438305  0.8693974   0.815725    0.75390226  0.6845467
  0.60835147  0.52607775  0.43854767  0.3466353   0.25125992  0.15337405
  0.05395571 -0.04600174 -0.14550003 -0.24354452 -0.3391547  -0.43137702
 -0.5192883  -0.6020119  -0.67872036 -0.7486465  -0.8110931  -0.8654355
 -0.91113025 -0.9477217  -0.97484356 -0.99222535 -0.99969304 -0.9971722
 -0.9846878  -0.9623649  -0.9304262  -0.8891913 ]

结果