Using automatic differentiation libraries to compute partial derivatives of an arbitrary tensor
(Note: this is not a question about back-propagation.) 
I am trying so solve on a GPU a non-linear PDE using PyTorch tensors in place of Numpy arrays. I want to calculate the partial derivatives of an arbitrary tensor, akin to the action of the center finite-difference numpy.gradient function. I have other ways around this problem, but since I am already using PyTorch, I'm wondering if it is possible use the autograd module (or, in general, any other autodifferentiation module) to perform this action.

I have created a tensor-compatible version of the numpy.gradient function - which runs a lot faster. But perhaps there is a more elegant way of doing this. I can't find any other sources that address this question, either to show that it's possible or impossible; perhaps this reflects my ignorance with the autodifferentiation algorithms.

I've had this same question myself: when numerically solving PDEs, we need access to spatial gradients (which the numpy.gradients function can give us) all the time - could it be possible to use automatic differentiation to compute the gradients, instead of using finite-difference or some flavor of it?

"I'm wondering if it is possible use the autograd module (or, in general, any other autodifferentiation module) to perform this action."

The answer is no: as soon as you discretize your problem in space or time, then time and space become discrete variables with a grid-like structure, and are not explicit variables which you feed into some function to compute the solution to the PDE. 

For example, if I wanted to compute the velocity field of some fluid flow u(x,t), I would discretize in space and time, and I would have u[:,:] where the indices represent positions in space and time. 

Automatic differentiation can compute the derivative of a function u(x,t). So why can't it compute the spatial or time derivative here? Because you've discretized your problem. This means you don't have a function for u for arbitrary x, but rather a function of u at some grid points. You can't differentiate automatically with respect to the spacing of the grid points.

As far as I can tell, the tensor-compatible function you've written is probably your best bet. You can see that a similar question has been asked in the PyTorch forums here and here. Or you could do something like 

dx = x[:,:,1:]-x[:,:,:-1]

if you're not worried about the endpoints. 
