This is an old-fashion technique : Looking at the hardware resources , eg: Memory /processing speed/floating/fixed point (for precision, convergence ...)  , mostly importantly the software /your code demands in terms of complexity (time /space) . 

With NN the neuron # , depth /layers makes it pretty expensive .  That is need for not only code fitting but intermediate buffers for storage is a huge demand.

The question is can you fit all this along with housekeeping software entities ...

Depends on your requirements/intentions:

If you want to use the Arduino for teaching purposes or simple/not-too-fast neuronal networks, this should be possible. (This is to some extend how NNs implementations on microcontrollers started some 20 years ago.)

Do not expect too much regarding the possible complexity and performance of such an NN...

If you want to dive into more recent NNs, a PC-based development seems more adequate. This allows for much better insight into the operation as well as debugging.

Regards

If your problem involves very less number of codings, the Arduino UNO/MEGA may be useful. Some examples and related work pertaining to the Arduino UNO with neural network is presented in the following links.

http://robotics.hobbizine.com/arduinoann.html

https://books.google.co.in/books?id=_Pi5BQAAQBAJ&pg=PA86&lpg=PA86&dq=Is+that+we+can+use+an+Arduino+to+incorporate+a+neural+network+program+for+practical+use?&source=bl&ots=wgJZbB6Fi8&sig=Kluq4BSo7i954hdJ338BsxLIEQA&hl=en&sa=X&ved=0ahUKEwjQ5vjoyJPRAhVLNo8KHQD4B3oQ6AEIOTAF#v=onepage&q=Is%20that%20we%20can%20use%20an%20Arduino%20to%20incorporate%20a%20neural%20network%20program%20for%20practical%20use%3F&f=false

http://arduinobasics.blogspot.in/2011/08/neural-network-part-1-connection.html

http://robotics.hobbizine.com/arduinoann.html

it's Ok, thank you very much dear  colleagues