Stochastic Gradient Descent with integer arithmetics
see eg Neural network training with constrained integer weights
Plagianakos, V.P.;   Vrahatis, M.N.;  

An Integer Recurrent Artificial Neural Network
for Classifying Feature Vectors 
Roelof K Brouwer PEng, Ph

the general idea in various implementations is to represent/approximate real numbers $x \in[0..1]$ as integers in the form $x \cdot 10^m$ where $m$ is some integer exponent.