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What are some of the shortcomings of a Neural Network?

It appears to be very good at pattern recognition, and uses a statistical analysis to rank the objects.

But can it synthesize? And create new knowledge, new science, new math, new techniques, new processes?

Well, today no neural networks can't do things like creating new knowledge. I encourage you, however, to check out GAN's, Neural Networks that can produce new and original results, that is to say, they can generate faces and digits, which do not exist. Also, there are neural networks capable of producing music, poetry and using style transfer they are able to transfer painting styles. So, I think that the future looks promising for the area of Neural Networks, but there is still a lot of work to do

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