Does Tessaract OCR uses neural networks as their default training mechanism

Yes, starting from tesseract 4.0, it provides a new lstm-based ocr engine: https://tesseract-ocr.github.io/tessdoc/NeuralNetsInTesseract4.00

It appears that Tessaract uses an Adaptive Classifier by default. Check this out for a good read:

https://github.com/tesseract-ocr/docs/blob/master/tesseracticdar2007.pdf

There appears to be an option called "Cube mode" where it will switch to using NNs for the learning system instead of the adaptive classifier (https://code.google.com/p/tesseract-ocr-extradocs/wiki/Cube). More info about adaptive classifiers:

http://www.cs.indiana.edu/~rawlins/website/adaptivity/information-helper.html

Also, related very closely is a Learning Classifier System:

http://en.wikipedia.org/wiki/Learning_classifier_system

Also, your terminology of "training" is very close. Training is how you teach the pattern recognition system or learning system what responses it should give to certain input sets. Then, it uses similarities when it encounters unknown data to classify the new data. Machine learning is one of the coolest fields in existence in my opinion (probably biased opinion but whatever!) keep up the learning! You are the meta learner: learning how to teach a machine to learn! Cool stuff!