Numpy 使用np.einsum和np.tensordot执行四阶张量的坐标变换

方程式是

$C'_{ijkl} = Q_{im} Q_{jn} C_{mnop} (Q^{-1})_{ok} (Q^{-1})_{pl}$

我能够使用

np.einsum('im,jn,mnop,ok,pl', Q, Q, C, Q_inv, Q_inv)

做这项工作，也期望

np.tensordot(np.tensordot(np.tensordot(Q, np.tensordot(Q, C, axes=[1,1]), axes=[1,0]), Q_inv, axes=[2,0]), Q_inv, axes=[3,0])

去工作，但是没有

具体内容：

C是一个四阶弹性张量：

array([[[[ 552.62389047,   -0.28689554,   -0.32194701],
         [  -0.28689554,  118.89168597,   -0.65559912],
         [  -0.32194701,   -0.65559912,  130.21758722]],

        [[  -0.28689554,  166.02923119,   -0.00000123],
         [ 166.02923119,    0.49494431,   -0.00000127],
         [  -0.00000123,   -0.00000127,   -0.57156702]],

        [[  -0.32194701,   -0.00000123,  165.99413061],
         [  -0.00000123,   -0.64666809,   -0.0000013 ],
         [ 165.99413061,   -0.0000013 ,    0.42997465]]],


       [[[  -0.28689554,  166.02923119,   -0.00000123],
         [ 166.02923119,    0.49494431,   -0.00000127],
         [  -0.00000123,   -0.00000127,   -0.57156702]],

        [[ 118.89168597,    0.49494431,   -0.64666809],
         [   0.49494431,  516.15898907,   -0.33132485],
         [  -0.64666809,   -0.33132485,  140.09010389]],

        [[  -0.65559912,   -0.00000127,   -0.0000013 ],
         [  -0.00000127,   -0.33132485,  165.98553869],
         [  -0.0000013 ,  165.98553869,    0.41913346]]],


       [[[  -0.32194701,   -0.00000123,  165.99413061],
         [  -0.00000123,   -0.64666809,   -0.0000013 ],
         [ 165.99413061,   -0.0000013 ,    0.42997465]],

        [[  -0.65559912,   -0.00000127,   -0.0000013 ],
         [  -0.00000127,   -0.33132485,  165.98553869],
         [  -0.0000013 ,  165.98553869,    0.41913346]],

        [[ 130.21758722,   -0.57156702,    0.42997465],
         [  -0.57156702,  140.09010389,    0.41913346],
         [   0.42997465,    0.41913346,  486.62412063]]]])

Q是改变x和y坐标的旋转矩阵

array([[ 0,  1,  0],
       [-1,  0,  0],
       [ 0,  0,  1]])

Q_inv是

array([[-0., -1., -0.],
       [ 1.,  0.,  0.],
       [ 0.,  0.,  1.]])

np.einsum

导致

array([[[[ 516.15898907,   -0.49494431,   -0.33132485],
         [  -0.49494431,  118.89168597,    0.64666809],
         [  -0.33132485,    0.64666809,  140.09010389]],

        [[  -0.49494431,  166.02923119,    0.00000127],
         [ 166.02923119,    0.28689554,   -0.00000123],
         [   0.00000127,   -0.00000123,    0.57156702]],

        [[  -0.33132485,    0.00000127,  165.98553869],
         [   0.00000127,   -0.65559912,    0.0000013 ],
         [ 165.98553869,    0.0000013 ,    0.41913346]]],


       [[[  -0.49494431,  166.02923119,    0.00000127],
         [ 166.02923119,    0.28689554,   -0.00000123],
         [   0.00000127,   -0.00000123,    0.57156702]],

        [[ 118.89168597,    0.28689554,   -0.65559912],
         [   0.28689554,  552.62389047,    0.32194701],
         [  -0.65559912,    0.32194701,  130.21758722]],

        [[   0.64666809,   -0.00000123,    0.0000013 ],
         [  -0.00000123,    0.32194701,  165.99413061],
         [   0.0000013 ,  165.99413061,   -0.42997465]]],


       [[[  -0.33132485,    0.00000127,  165.98553869],
         [   0.00000127,   -0.65559912,    0.0000013 ],
         [ 165.98553869,    0.0000013 ,    0.41913346]],

        [[   0.64666809,   -0.00000123,    0.0000013 ],
         [  -0.00000123,    0.32194701,  165.99413061],
         [   0.0000013 ,  165.99413061,   -0.42997465]],

        [[ 140.09010389,    0.57156702,    0.41913346],
         [   0.57156702,  130.21758722,   -0.42997465],
         [   0.41913346,   -0.42997465,  486.62412063]]]])

array([[[[ 552.62389047,   -0.28689554,    0.32194701],
         [  -0.28689554,  118.89168597,    0.65559912],
         [  -0.32194701,   -0.65559912, -130.21758722]],

        [[  -0.28689554,  166.02923119,    0.00000123],
         [ 166.02923119,    0.49494431,    0.00000127],
         [  -0.00000123,   -0.00000127,    0.57156702]],

        [[  -0.32194701,   -0.00000123, -165.99413061],
         [  -0.00000123,   -0.64666809,    0.0000013 ],
         [ 165.99413061,   -0.0000013 ,   -0.42997465]]],


       [[[  -0.28689554,  166.02923119,    0.00000123],
         [ 166.02923119,    0.49494431,    0.00000127],
         [  -0.00000123,   -0.00000127,    0.57156702]],

        [[ 118.89168597,    0.49494431,    0.64666809],
         [   0.49494431,  516.15898907,    0.33132485],
         [  -0.64666809,   -0.33132485, -140.09010389]],

        [[  -0.65559912,   -0.00000127,    0.0000013 ],
         [  -0.00000127,   -0.33132485, -165.98553869],
         [  -0.0000013 ,  165.98553869,   -0.41913346]]],


       [[[   0.32194701,    0.00000123,  165.99413061],
         [   0.00000123,    0.64666809,   -0.0000013 ],
         [-165.99413061,    0.0000013 ,    0.42997465]],

        [[   0.65559912,    0.00000127,   -0.0000013 ],
         [   0.00000127,    0.33132485,  165.98553869],
         [   0.0000013 , -165.98553869,    0.41913346]],

        [[-130.21758722,    0.57156702,    0.42997465],
         [   0.57156702, -140.09010389,    0.41913346],
         [  -0.42997465,   -0.41913346,  486.62412063]]]])

我认为这是正确的，而四个

array([[[[ 516.15898907,   -0.49494431,   -0.33132485],
         [  -0.49494431,  118.89168597,    0.64666809],
         [  -0.33132485,    0.64666809,  140.09010389]],

        [[  -0.49494431,  166.02923119,    0.00000127],
         [ 166.02923119,    0.28689554,   -0.00000123],
         [   0.00000127,   -0.00000123,    0.57156702]],

        [[  -0.33132485,    0.00000127,  165.98553869],
         [   0.00000127,   -0.65559912,    0.0000013 ],
         [ 165.98553869,    0.0000013 ,    0.41913346]]],


       [[[  -0.49494431,  166.02923119,    0.00000127],
         [ 166.02923119,    0.28689554,   -0.00000123],
         [   0.00000127,   -0.00000123,    0.57156702]],

        [[ 118.89168597,    0.28689554,   -0.65559912],
         [   0.28689554,  552.62389047,    0.32194701],
         [  -0.65559912,    0.32194701,  130.21758722]],

        [[   0.64666809,   -0.00000123,    0.0000013 ],
         [  -0.00000123,    0.32194701,  165.99413061],
         [   0.0000013 ,  165.99413061,   -0.42997465]]],


       [[[  -0.33132485,    0.00000127,  165.98553869],
         [   0.00000127,   -0.65559912,    0.0000013 ],
         [ 165.98553869,    0.0000013 ,    0.41913346]],

        [[   0.64666809,   -0.00000123,    0.0000013 ],
         [  -0.00000123,    0.32194701,  165.99413061],
         [   0.0000013 ,  165.99413061,   -0.42997465]],

        [[ 140.09010389,    0.57156702,    0.41913346],
         [   0.57156702,  130.21758722,   -0.42997465],
         [   0.41913346,   -0.42997465,  486.62412063]]]])

array([[[[ 552.62389047,   -0.28689554,    0.32194701],
         [  -0.28689554,  118.89168597,    0.65559912],
         [  -0.32194701,   -0.65559912, -130.21758722]],

        [[  -0.28689554,  166.02923119,    0.00000123],
         [ 166.02923119,    0.49494431,    0.00000127],
         [  -0.00000123,   -0.00000127,    0.57156702]],

        [[  -0.32194701,   -0.00000123, -165.99413061],
         [  -0.00000123,   -0.64666809,    0.0000013 ],
         [ 165.99413061,   -0.0000013 ,   -0.42997465]]],


       [[[  -0.28689554,  166.02923119,    0.00000123],
         [ 166.02923119,    0.49494431,    0.00000127],
         [  -0.00000123,   -0.00000127,    0.57156702]],

        [[ 118.89168597,    0.49494431,    0.64666809],
         [   0.49494431,  516.15898907,    0.33132485],
         [  -0.64666809,   -0.33132485, -140.09010389]],

        [[  -0.65559912,   -0.00000127,    0.0000013 ],
         [  -0.00000127,   -0.33132485, -165.98553869],
         [  -0.0000013 ,  165.98553869,   -0.41913346]]],


       [[[   0.32194701,    0.00000123,  165.99413061],
         [   0.00000123,    0.64666809,   -0.0000013 ],
         [-165.99413061,    0.0000013 ,    0.42997465]],

        [[   0.65559912,    0.00000127,   -0.0000013 ],
         [   0.00000127,    0.33132485,  165.98553869],
         [   0.0000013 , -165.98553869,    0.41913346]],

        [[-130.21758722,    0.57156702,    0.42997465],
         [   0.57156702, -140.09010389,    0.41913346],
         [  -0.42997465,   -0.41913346,  486.62412063]]]])

注意负的大数字。

方法#1

一种方法是使用

np.einsum

获得相同的结果，尽管不是在一个步骤中，而是在信任者的帮助下-

方法#2

如果您希望避免

广播

，而使用另外两个

np.tensordot

实例，您可以这样做-

# Perform "np.einsum('jn,mnop', Q, C). Notice how, Q is represented by 'jn' 
# and C by 'mnop'. We need to reduce the 'm' dimension, i.e. reduce 'axes=1'
# from Q and `axes=1` from C corresponding to `n' in each of the inputs.
# Thus, 'jn' + 'mnop' => 'jmop' after 'n' is reduced and order is maintained.
Q_C1 = np.tensordot(Q,C,axes=([1],[1]))

# Perform "np.einsum('im,jn,mnop', Q, Q, C). We need to use Q and Q_C1.
# Q is 'im' and Q_C1 is 'jmop'. Thus, again we need to reduce 'axes=1'
# from Q and `axes=1` from Q_C1 corresponding to `m' in each of the inputs.
# Thus, 'im' + 'jmop' => 'ijop' after 'm' is reduced and order is maintained.
parte1 = np.tensordot(Q,Q_C1,axes=([1],[1]))

# Use the same philosophy to get the rest of the einsum equivalent, 
# but use parte1 and go right and use Q_inv 
out = np.tensordot(np.tensordot(parte1,Q_inv,axes=([2],[0])),Q_inv,axes=([2],[0]))

---

np.tensordot

的诀窍是跟踪被

轴

参数缩小的尺寸，以及折叠的尺寸如何与其余输入的尺寸对齐。

你的问题也行不通；这是不完整的。告诉我们tensordot在哪里/如何工作-错误和/或回答错误（特别是形状）。对此表示抱歉。我想调用

np.tensordot

时一定有一些明显的错误，只要看一下就很容易发现。它更新了一个例子。你（或某人）需要一步一步地调试它。换句话说，确保每个

tensordot

与等效的

einsum

匹配。4个嵌套的

tensordot

表达式很难让我集中精力。它很有效。但是

Q[：，None，：，None]*Q[：，None，：]

和

Q[：，None，：，None]*Q[None，：，None，：]

一样吗？我仍然很好奇为什么四个

tensordot

解决方案不起作用。你能举例说明吗？@terencezl没错。为简洁起见，省略了前导的

None

，但为了更好地理解，您可能希望保留它。它确实有意义。但是

np.tensordot（Q，np.tensordot（Q，C，axes=[1,1]），axes=[1,0]）

对上半部分也做了两步缩减：

np.einsum（'im，jn，mnop'，Q，Q，C）

？它们给出了不同的结果。我想这就是问题和我的困惑所在。看起来是

np。tensordot（Q，np.tensordot（Q，C，axes=[1,0]），axes=[1,1]）

是正确的方法。请注意轴的顺序。为什么它们不同？