If you're implementing a feedforward network from scratch, e.g. in Python with something like numpy, you can use the stored weight matrix from previous training and just update it with further back propagation steps. Here is a good tutorial:

https://databoys.github.io/Feedforward/

You would just be updating the 'w' arrays. 

Hi Payam, 

thank you for your question. I recommend to use matlab/octave if you are interest just in NN behaviour analysis - you can use a pipe (file) to feed your network with the data flow.

However, if you are coding something that you would like to integrate further, numpy cited by Mr. Zeldes seems the better option. 

Best wishes,

Fábio

Yes, if you want 'out of the box' solutions there are many options, in matlab, R, Java etc. If you're using Python I recommend TensorFlow: https://www.tensorflow.org/. I'm not sure how easy it is to continue training of an existing resource though, unless it's also in the same framework.

You can implement a model in R ,the link to fit a neural network is

https://www.r-bloggers.com/fitting-a-neural-network-in-r-neuralnet-package/

Once the network is trained and output is predicted ,you can store the weights of the previous training and update them upon the new training .However ,care should be taken not to over-fit the network.

Besides neuralnet package, R has various other packages like nnet,pcaNet. You would have to use them according to your application( that is whether you want to classify or predict the output) 

narx function in MATLAB