Really interesting, I have been confused about the einsum function before. As a former physicist, I would also like to see the actual tensor notation for the examples. So instead of ij,jk something like $A_i^j B_j^k$ (imagine the math here instead of the LaTeX).
Tensors used in deep learning are not the same as the definition used by Physicists - blame the DL community for this :). So DL tensors are just N-dimensional arrays of data, and there is no concept of covariance and contravariance of the dimensions. You could think of DL tensors as Cartesian tensors and they don't need to conform to the same transformation laws that Physics tensors do.
Einsum immediately clicked with me because in my past advanced classical mechanics courses such concise contractions of multi-index creatures were really the only way to make quick sense of complicated problems without the limitations of low-dimensional array notations. Physics tensors are of course different creatures than the simpler multidimensional arrays of PyTorch and co., but the simplified einsum notation still works very well. I ended up sometimes rewriting my Einsum code to plain tensor manipulation code in order to better work with collaborators who didnt like einsum, but I still experiment in my own hacks with einsum when I need to. Occasionally I felt that it would have been nice to also have the Levi-Civita symbol available in einsum, or to be able to use complex numbers and take complex conjugates, but these are all super-specialized requests and there often is a good way around them without modifying einsum.
This is a good idea, though one problem is that Einsum notation (as realized in Numpy and Pytorch) doesn't support the notion of co-contravariance, and the site is based on their Einsum notation. I could potentially add the variances for the examples, though that would move away from how the site currently works (where the information about the reduction comes only from the einsum input).
