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tf.nn.bidirectional_dynamic_rnn详

tf.nn.bidirectional_dynamic_rnn详

作者: 全村希望gone | 来源:发表于2019-06-24 17:46 被阅读0次

这个函数我已经查了三遍不止了,每次看网上说的那些都没看明白。也怪我自己不动脑子,现成的代码我复制粘贴了,连改都不改这怎么能懂其中的意思呢。我一开始看了这篇博客,觉得写得还真好,详细解读了内部,什么两次reverse后再拼接啥的(如下图)

让我以为做反向lstm的时候先把输入A逆序排列一下,进入dynamic_rnn后得到反向输出B,再将B逆序排列一下,得到最终的C,最终这个函数输出的就是正向lstm值和“反向lstm值C”。我当时就就在想两次reverse后,这个反向lstm输出的值不就相当于还是从左往右的信息嘛,带着这种观点过了好多天,知道今天我运行了别人的代码发现并不是这样。(事实证明,这个函数就是这样的!!!有两个例子,第二个例子更清楚)。
理解bilstm的关键在于反向lstm并不是像看起来那样从右往左传递信息,而是先将原来的输入逆序排列输入到正向lstm中,再将得到的输出结果逆序排列,便得到了所谓的“反向lstm”的输出

例一

import tensorflow as tf
import numpy as np

# 创建一个batch为2的三维数组,值全为1
X = np.ones((2, 10, 8))
# 指定每个batch的真实长度,这是bidirectional_dynamic_rnn中的一个参数,如果不指定,默认为batch的最大长度
X_lengths = [10, 10]

cell = tf.nn.rnn_cell.LSTMCell(num_units=20, state_is_tuple=True)

outputs, states = tf.nn.bidirectional_dynamic_rnn(cell_fw=cell, cell_bw=cell, dtype=tf.float64,
                                                  sequence_length=X_lengths, inputs=X)

# bidirectional_dynamic_rnn输出两个元组,第一个元组为输出值元组,第二个为状态元组
output_fw, output_bw = outputs
states_fw, states_bw = states

with tf.Session() as sess:
    sess.run(tf.global_variables_initializer())
    # 状态元组中的每个元素也还是一个元组,包含两个元素c和h,c是hidden state,h是output state
    c_fw, h_fw = states_fw
    c_bw, h_bw = states_bw
    print('***********print h_fw')
    print(sess.run(h_fw))
    print('***********print h_bw')
    print(sess.run(h_bw))
    print('***********print c_fw')
    print(sess.run(c_fw))
    print('***********print c_bw')
    print(sess.run(c_bw))
    print('***********print output_fw')
    print(sess.run(output_fw))
    print('***********print output_bw')
    print(sess.run(output_bw))

输出结果如下

***********print h_fw
[[ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248 -0.37795411
  -0.38890486  0.24909768  0.24172305 -0.27529398 -0.18520042 -0.39698435
   0.0636031  -0.26719873  0.18002771  0.09966553  0.02702067 -0.29513662
  -0.09883436 -0.41679873]
 [ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248 -0.37795411
  -0.38890486  0.24909768  0.24172305 -0.27529398 -0.18520042 -0.39698435
   0.0636031  -0.26719873  0.18002771  0.09966553  0.02702067 -0.29513662
  -0.09883436 -0.41679873]]
***********print h_bw
[[ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248 -0.37795411
  -0.38890486  0.24909768  0.24172305 -0.27529398 -0.18520042 -0.39698435
   0.0636031  -0.26719873  0.18002771  0.09966553  0.02702067 -0.29513662
  -0.09883436 -0.41679873]
 [ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248 -0.37795411
  -0.38890486  0.24909768  0.24172305 -0.27529398 -0.18520042 -0.39698435
   0.0636031  -0.26719873  0.18002771  0.09966553  0.02702067 -0.29513662
  -0.09883436 -0.41679873]]
***********print c_fw
[[ 0.63050086 -0.14962187  0.17986396 -0.27891692 -0.83289912 -1.38633692
  -0.92382506  0.47125799  0.71944087 -0.55796188 -0.4651182  -1.34933772
   0.1275259  -0.79503812  0.44752065  0.23413849  0.04473469 -0.82337067
  -0.20446274 -0.80014184]
 [ 0.63050086 -0.14962187  0.17986396 -0.27891692 -0.83289912 -1.38633692
  -0.92382506  0.47125799  0.71944087 -0.55796188 -0.4651182  -1.34933772
   0.1275259  -0.79503812  0.44752065  0.23413849  0.04473469 -0.82337067
  -0.20446274 -0.80014184]]
***********print c_bw
[[ 0.63050086 -0.14962187  0.17986396 -0.27891692 -0.83289912 -1.38633692
  -0.92382506  0.47125799  0.71944087 -0.55796188 -0.4651182  -1.34933772
   0.1275259  -0.79503812  0.44752065  0.23413849  0.04473469 -0.82337067
  -0.20446274 -0.80014184]
 [ 0.63050086 -0.14962187  0.17986396 -0.27891692 -0.83289912 -1.38633692
  -0.92382506  0.47125799  0.71944087 -0.55796188 -0.4651182  -1.34933772
   0.1275259  -0.79503812  0.44752065  0.23413849  0.04473469 -0.82337067
  -0.20446274 -0.80014184]]
***********print output_fw
[[[ 0.09213119 -0.00920132 -0.00480822 -0.06614824 -0.10940994
   -0.13237866 -0.06281013  0.08518363  0.07377088 -0.00439786
   -0.03767817 -0.1139195   0.01487621 -0.05797102  0.09557128
    0.00161788 -0.06577087 -0.08184791 -0.05674727 -0.22975524]
  [ 0.15855342 -0.02402754 -0.00301006 -0.09785019 -0.17732144
   -0.22491602 -0.1349754   0.13963072  0.12761351 -0.02347015
   -0.07433172 -0.19753812  0.02575886 -0.11248398  0.15490154
    0.0149613  -0.10369941 -0.14653386 -0.08158803 -0.33888571]
  [ 0.20518182 -0.03977035  0.00220437 -0.11144359 -0.21780033
   -0.28412027 -0.20131872  0.17511601  0.16677614 -0.05516816
   -0.10460625 -0.25808741  0.03480547 -0.15743644  0.18638093
    0.03129929 -0.11566092 -0.19303162 -0.09320168 -0.38787705]
  [ 0.23793991 -0.0540012   0.00920335 -0.11647142 -0.24172754
   -0.32073244 -0.25595129  0.19880824  0.19462017 -0.09352138
   -0.127896   -0.30146283  0.04269355 -0.1917475   0.20078535
    0.04700247 -0.10889583 -0.22526698 -0.09881287 -0.40967251]
  [ 0.26108442 -0.06564608  0.0171266  -0.1179926  -0.25581146
   -0.34330216 -0.2981817   0.21500322  0.21370547 -0.13296083
   -0.14530196 -0.33239821  0.04941772 -0.21687859  0.20531662
    0.06073976 -0.09068629 -0.24765488 -0.10130358 -0.41883299]
  [ 0.27751227 -0.07446752  0.02544216 -0.11856258 -0.26406581
   -0.35734935 -0.32964097  0.22637017  0.22625771 -0.16993574
   -0.15821977 -0.35452075  0.05483244 -0.23490698  0.20414891
    0.07220202 -0.06670979 -0.26350145 -0.10203061 -0.42187412]
  [ 0.28920624 -0.08068884  0.03378522 -0.1193405  -0.26889405
   -0.36621415 -0.35256554  0.23458705  0.2341066  -0.20278181
   -0.1678622  -0.37046562  0.05886875 -0.24768901  0.19974503
    0.08152418 -0.04094146 -0.27499945 -0.10175494 -0.4219268 ]
  [ 0.2975349  -0.08473076  0.04189705 -0.12071914 -0.27173138
   -0.37188732 -0.3690315   0.24071022  0.2386569  -0.23108288
   -0.17515872 -0.3820775   0.061575   -0.25666779  0.19364629
    0.08900016 -0.01591296 -0.28355518 -0.10094973 -0.42059815]
  [ 0.30344973 -0.08705629  0.04959632 -0.12270135 -0.27342634
   -0.37555987 -0.38071883  0.24540596  0.24094529 -0.25508262
   -0.18078124 -0.39062611  0.06309015 -0.26291155  0.18685659
    0.09494983  0.0069545  -0.29007129 -0.09991513 -0.41874011]
  [ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248
   -0.37795411 -0.38890486  0.24909768  0.24172305 -0.27529398
   -0.18520042 -0.39698435  0.0636031  -0.26719873  0.18002771
    0.09966553  0.02702067 -0.29513662 -0.09883436 -0.41679873]]

 [[ 0.09213119 -0.00920132 -0.00480822 -0.06614824 -0.10940994
   -0.13237866 -0.06281013  0.08518363  0.07377088 -0.00439786
   -0.03767817 -0.1139195   0.01487621 -0.05797102  0.09557128
    0.00161788 -0.06577087 -0.08184791 -0.05674727 -0.22975524]
  [ 0.15855342 -0.02402754 -0.00301006 -0.09785019 -0.17732144
   -0.22491602 -0.1349754   0.13963072  0.12761351 -0.02347015
   -0.07433172 -0.19753812  0.02575886 -0.11248398  0.15490154
    0.0149613  -0.10369941 -0.14653386 -0.08158803 -0.33888571]
  [ 0.20518182 -0.03977035  0.00220437 -0.11144359 -0.21780033
   -0.28412027 -0.20131872  0.17511601  0.16677614 -0.05516816
   -0.10460625 -0.25808741  0.03480547 -0.15743644  0.18638093
    0.03129929 -0.11566092 -0.19303162 -0.09320168 -0.38787705]
  [ 0.23793991 -0.0540012   0.00920335 -0.11647142 -0.24172754
   -0.32073244 -0.25595129  0.19880824  0.19462017 -0.09352138
   -0.127896   -0.30146283  0.04269355 -0.1917475   0.20078535
    0.04700247 -0.10889583 -0.22526698 -0.09881287 -0.40967251]
  [ 0.26108442 -0.06564608  0.0171266  -0.1179926  -0.25581146
   -0.34330216 -0.2981817   0.21500322  0.21370547 -0.13296083
   -0.14530196 -0.33239821  0.04941772 -0.21687859  0.20531662
    0.06073976 -0.09068629 -0.24765488 -0.10130358 -0.41883299]
  [ 0.27751227 -0.07446752  0.02544216 -0.11856258 -0.26406581
   -0.35734935 -0.32964097  0.22637017  0.22625771 -0.16993574
   -0.15821977 -0.35452075  0.05483244 -0.23490698  0.20414891
    0.07220202 -0.06670979 -0.26350145 -0.10203061 -0.42187412]
  [ 0.28920624 -0.08068884  0.03378522 -0.1193405  -0.26889405
   -0.36621415 -0.35256554  0.23458705  0.2341066  -0.20278181
   -0.1678622  -0.37046562  0.05886875 -0.24768901  0.19974503
    0.08152418 -0.04094146 -0.27499945 -0.10175494 -0.4219268 ]
  [ 0.2975349  -0.08473076  0.04189705 -0.12071914 -0.27173138
   -0.37188732 -0.3690315   0.24071022  0.2386569  -0.23108288
   -0.17515872 -0.3820775   0.061575   -0.25666779  0.19364629
    0.08900016 -0.01591296 -0.28355518 -0.10094973 -0.42059815]
  [ 0.30344973 -0.08705629  0.04959632 -0.12270135 -0.27342634
   -0.37555987 -0.38071883  0.24540596  0.24094529 -0.25508262
   -0.18078124 -0.39062611  0.06309015 -0.26291155  0.18685659
    0.09494983  0.0069545  -0.29007129 -0.09991513 -0.41874011]
  [ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248
   -0.37795411 -0.38890486  0.24909768  0.24172305 -0.27529398
   -0.18520042 -0.39698435  0.0636031  -0.26719873  0.18002771
    0.09966553  0.02702067 -0.29513662 -0.09883436 -0.41679873]]]
***********print output_bw
[[[ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248
   -0.37795411 -0.38890486  0.24909768  0.24172305 -0.27529398
   -0.18520042 -0.39698435  0.0636031  -0.26719873  0.18002771
    0.09966553  0.02702067 -0.29513662 -0.09883436 -0.41679873]
  [ 0.30344973 -0.08705629  0.04959632 -0.12270135 -0.27342634
   -0.37555987 -0.38071883  0.24540596  0.24094529 -0.25508262
   -0.18078124 -0.39062611  0.06309015 -0.26291155  0.18685659
    0.09494983  0.0069545  -0.29007129 -0.09991513 -0.41874011]
  [ 0.2975349  -0.08473076  0.04189705 -0.12071914 -0.27173138
   -0.37188732 -0.3690315   0.24071022  0.2386569  -0.23108288
   -0.17515872 -0.3820775   0.061575   -0.25666779  0.19364629
    0.08900016 -0.01591296 -0.28355518 -0.10094973 -0.42059815]
  [ 0.28920624 -0.08068884  0.03378522 -0.1193405  -0.26889405
   -0.36621415 -0.35256554  0.23458705  0.2341066  -0.20278181
   -0.1678622  -0.37046562  0.05886875 -0.24768901  0.19974503
    0.08152418 -0.04094146 -0.27499945 -0.10175494 -0.4219268 ]
  [ 0.27751227 -0.07446752  0.02544216 -0.11856258 -0.26406581
   -0.35734935 -0.32964097  0.22637017  0.22625771 -0.16993574
   -0.15821977 -0.35452075  0.05483244 -0.23490698  0.20414891
    0.07220202 -0.06670979 -0.26350145 -0.10203061 -0.42187412]
  [ 0.26108442 -0.06564608  0.0171266  -0.1179926  -0.25581146
   -0.34330216 -0.2981817   0.21500322  0.21370547 -0.13296083
   -0.14530196 -0.33239821  0.04941772 -0.21687859  0.20531662
    0.06073976 -0.09068629 -0.24765488 -0.10130358 -0.41883299]
  [ 0.23793991 -0.0540012   0.00920335 -0.11647142 -0.24172754
   -0.32073244 -0.25595129  0.19880824  0.19462017 -0.09352138
   -0.127896   -0.30146283  0.04269355 -0.1917475   0.20078535
    0.04700247 -0.10889583 -0.22526698 -0.09881287 -0.40967251]
  [ 0.20518182 -0.03977035  0.00220437 -0.11144359 -0.21780033
   -0.28412027 -0.20131872  0.17511601  0.16677614 -0.05516816
   -0.10460625 -0.25808741  0.03480547 -0.15743644  0.18638093
    0.03129929 -0.11566092 -0.19303162 -0.09320168 -0.38787705]
  [ 0.15855342 -0.02402754 -0.00301006 -0.09785019 -0.17732144
   -0.22491602 -0.1349754   0.13963072  0.12761351 -0.02347015
   -0.07433172 -0.19753812  0.02575886 -0.11248398  0.15490154
    0.0149613  -0.10369941 -0.14653386 -0.08158803 -0.33888571]
  [ 0.09213119 -0.00920132 -0.00480822 -0.06614824 -0.10940994
   -0.13237866 -0.06281013  0.08518363  0.07377088 -0.00439786
   -0.03767817 -0.1139195   0.01487621 -0.05797102  0.09557128
    0.00161788 -0.06577087 -0.08184791 -0.05674727 -0.22975524]]

 [[ 0.30761863 -0.08809378  0.05676232 -0.12512271 -0.27447248
   -0.37795411 -0.38890486  0.24909768  0.24172305 -0.27529398
   -0.18520042 -0.39698435  0.0636031  -0.26719873  0.18002771
    0.09966553  0.02702067 -0.29513662 -0.09883436 -0.41679873]
  [ 0.30344973 -0.08705629  0.04959632 -0.12270135 -0.27342634
   -0.37555987 -0.38071883  0.24540596  0.24094529 -0.25508262
   -0.18078124 -0.39062611  0.06309015 -0.26291155  0.18685659
    0.09494983  0.0069545  -0.29007129 -0.09991513 -0.41874011]
  [ 0.2975349  -0.08473076  0.04189705 -0.12071914 -0.27173138
   -0.37188732 -0.3690315   0.24071022  0.2386569  -0.23108288
   -0.17515872 -0.3820775   0.061575   -0.25666779  0.19364629
    0.08900016 -0.01591296 -0.28355518 -0.10094973 -0.42059815]
  [ 0.28920624 -0.08068884  0.03378522 -0.1193405  -0.26889405
   -0.36621415 -0.35256554  0.23458705  0.2341066  -0.20278181
   -0.1678622  -0.37046562  0.05886875 -0.24768901  0.19974503
    0.08152418 -0.04094146 -0.27499945 -0.10175494 -0.4219268 ]
  [ 0.27751227 -0.07446752  0.02544216 -0.11856258 -0.26406581
   -0.35734935 -0.32964097  0.22637017  0.22625771 -0.16993574
   -0.15821977 -0.35452075  0.05483244 -0.23490698  0.20414891
    0.07220202 -0.06670979 -0.26350145 -0.10203061 -0.42187412]
  [ 0.26108442 -0.06564608  0.0171266  -0.1179926  -0.25581146
   -0.34330216 -0.2981817   0.21500322  0.21370547 -0.13296083
   -0.14530196 -0.33239821  0.04941772 -0.21687859  0.20531662
    0.06073976 -0.09068629 -0.24765488 -0.10130358 -0.41883299]
  [ 0.23793991 -0.0540012   0.00920335 -0.11647142 -0.24172754
   -0.32073244 -0.25595129  0.19880824  0.19462017 -0.09352138
   -0.127896   -0.30146283  0.04269355 -0.1917475   0.20078535
    0.04700247 -0.10889583 -0.22526698 -0.09881287 -0.40967251]
  [ 0.20518182 -0.03977035  0.00220437 -0.11144359 -0.21780033
   -0.28412027 -0.20131872  0.17511601  0.16677614 -0.05516816
   -0.10460625 -0.25808741  0.03480547 -0.15743644  0.18638093
    0.03129929 -0.11566092 -0.19303162 -0.09320168 -0.38787705]
  [ 0.15855342 -0.02402754 -0.00301006 -0.09785019 -0.17732144
   -0.22491602 -0.1349754   0.13963072  0.12761351 -0.02347015
   -0.07433172 -0.19753812  0.02575886 -0.11248398  0.15490154
    0.0149613  -0.10369941 -0.14653386 -0.08158803 -0.33888571]
  [ 0.09213119 -0.00920132 -0.00480822 -0.06614824 -0.10940994
   -0.13237866 -0.06281013  0.08518363  0.07377088 -0.00439786
   -0.03767817 -0.1139195   0.01487621 -0.05797102  0.09557128
    0.00161788 -0.06577087 -0.08184791 -0.05674727 -0.22975524]]]

从输出中可以看出以下几点

  • 每个batch的hidden state和output_state都只包含一个tensor,在正向lstm中,h_fw就是output的最后一个tensor,在反向lstm中,h_bw就是output的第一个tensor。hidden_state和output_state是同样的机制,它们都表示最后的状态。
  • 再看结果,可以看到output_bw和output_fw是完全相反的,如果只做一次reverse,应该是和原来相同(因为值全为1,所以reverse后还是和原来的输入相等)。

例二

直接上代码

import tensorflow as tf
import numpy as np

# 创建一个batch为2的三维数组
X = np.array([[[1, 2, 3, 4, 5], [2, 3, 4, 5, 6], [3, 4, 5, 6, 7]], [[3, 4, 5, 6, 7], [2, 3, 4, 5, 6], [1, 2, 3, 4, 5]]],
             dtype=np.float64)
# 指定每个batch的真实长度,这是bidirectional_dynamic_rnn中的一个参数,如果不指定,默认为batch的最大长度
X_lengths = [3, 3]

cell = tf.nn.rnn_cell.LSTMCell(num_units=10, state_is_tuple=True)

outputs, states = tf.nn.bidirectional_dynamic_rnn(cell_fw=cell, cell_bw=cell, dtype=tf.float64,
                                                  sequence_length=X_lengths, inputs=X)

# bidirectional_dynamic_rnn输出两个元组,第一个元组为输出值元组,第二个为状态元组
output_fw, output_bw = outputs
states_fw, states_bw = states

with tf.Session() as sess:
    sess.run(tf.global_variables_initializer())
    # 状态元组中的每个元素也还是一个元组,包含两个元素c和h,c是hidden state,h是output state
    c_fw, h_fw = states_fw
    c_bw, h_bw = states_bw
    print('***********print h_fw')
    print(sess.run(h_fw))
    print('***********print h_bw')
    print(sess.run(h_bw))
    print('***********print c_fw')
    print(sess.run(c_fw))
    print('***********print c_bw')
    print(sess.run(c_bw))
    print('***********print output_fw')
    print(sess.run(output_fw))
    print('***********print output_bw')
    print(sess.run(output_bw))

输出结果

***********print h_fw
[[-0.90904258 -0.72417344 -0.00340704  0.47999107 -0.6172267   0.16119028
   0.96314568  0.59865986 -0.23445085  0.14730728]
 [-0.80015177 -0.65880448 -0.01965226  0.42119995 -0.43309781  0.18892933
   0.90951544  0.43914896 -0.26462378  0.1727273 ]]
***********print h_bw
[[-0.80015177 -0.65880448 -0.01965226  0.42119995 -0.43309781  0.18892933
   0.90951544  0.43914896 -0.26462378  0.1727273 ]
 [-0.90904258 -0.72417344 -0.00340704  0.47999107 -0.6172267   0.16119028
   0.96314568  0.59865986 -0.23445085  0.14730728]]
***********print c_fw
[[-2.00750908 -1.19615351 -0.28910374  0.99731872 -1.66288234  1.35553407
   2.10854366  0.72069906 -0.27357135  0.9675592 ]
 [-1.97285668 -1.16508798 -0.28735088  1.00169352 -1.40325954  1.3543562
   1.94142731  0.52746365 -0.34427745  0.93367482]]
***********print c_bw
[[-1.97285668 -1.16508798 -0.28735088  1.00169352 -1.40325954  1.3543562
   1.94142731  0.52746365 -0.34427745  0.93367482]
 [-2.00750908 -1.19615351 -0.28910374  0.99731872 -1.66288234  1.35553407
   2.10854366  0.72069906 -0.27357135  0.9675592 ]]
***********print output_fw
[[[-0.52647366 -0.41398232 -0.01570638  0.16715023 -0.25770299
    0.11789769  0.60452984  0.19651944 -0.07048413  0.18267517]
  [-0.8034695  -0.62766681 -0.00827474  0.34252566 -0.47038781
    0.15523849  0.87603592  0.43703746 -0.16224657  0.18179719]
  [-0.90904258 -0.72417344 -0.00340704  0.47999107 -0.6172267
    0.16119028  0.96314568  0.59865986 -0.23445085  0.14730728]]

 [[-0.64887064 -0.49435156 -0.00098317  0.22632825 -0.45637361
    0.10613126  0.6985296   0.4481981  -0.04567536  0.16216273]
  [-0.8229839  -0.64132454 -0.00468272  0.3630846  -0.49435403
    0.15683851  0.89261432  0.51742868 -0.15192277  0.16823444]
  [-0.80015177 -0.65880448 -0.01965226  0.42119995 -0.43309781
    0.18892933  0.90951544  0.43914896 -0.26462378  0.1727273 ]]]
***********print output_bw
[[[-0.80015177 -0.65880448 -0.01965226  0.42119995 -0.43309781
    0.18892933  0.90951544  0.43914896 -0.26462378  0.1727273 ]
  [-0.8229839  -0.64132454 -0.00468272  0.3630846  -0.49435403
    0.15683851  0.89261432  0.51742868 -0.15192277  0.16823444]
  [-0.64887064 -0.49435156 -0.00098317  0.22632825 -0.45637361
    0.10613126  0.6985296   0.4481981  -0.04567536  0.16216273]]

 [[-0.90904258 -0.72417344 -0.00340704  0.47999107 -0.6172267
    0.16119028  0.96314568  0.59865986 -0.23445085  0.14730728]
  [-0.8034695  -0.62766681 -0.00827474  0.34252566 -0.47038781
    0.15523849  0.87603592  0.43703746 -0.16224657  0.18179719]
  [-0.52647366 -0.41398232 -0.01570638  0.16715023 -0.25770299
    0.11789769  0.60452984  0.19651944 -0.07048413  0.18267517]]]

注意看正向和反向的输出



图解(字难看,图也乱,但是还能将就看。)

参考资料

[1] https://blog.csdn.net/u012436149/article/details/71080601
[2] https://blog.csdn.net/qq_41424519/article/details/82112904

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