I can answer for the Python: Tensorflow is a good platform. It is developed by Google. So it has a high quality. Documentation is not ideal though, not so many examples there. But stackoverflow has lots of them.
My impression is, that for python, Keras might be a good starting point. For fortran there aren't a lot of choices. But the ones that B. M mentioned seems good, particularly fann.
It is worth checking out the language Julia. It is as easy as python, and so far at least for my atomistic Monte Carlo and Molecular Dynamics codes, is as fast as Fortran. When it is slower than Fortran the difference is measurable in percentage, never order of magnitudes. Also, I find it to be significantly faster than python + numba. However, a skilled user in python can make the python code faster I am sure, but I doubt it will compete with Julia. They have optimized it from the ground up for speed and scientific computing. The thing I like about Julia is it starts out fast, and a skilled user can make it even faster :)
As for ANN, this is out of my field but there seems to be more and more options for it in Julia.
We just released the Fortran Keras Bridge (FKB). It's a two-way bridge between ecosystems where deep learning resources are plentiful (Python) and those where they're scarce (Fortran). By leveraging FKB you can easily train networks in Python and transfer them to Fortran and vice versa.