  Many natural optimization problems are NP-hard, which implies that they are
probably hard to solve exactly in the worst-case. However, it suffices to get
reasonably good solutions for all (or even most) instances in practice. This
paper presents a new algorithm for computing approximate solutions in
${\Theta(N})$ for the Maximum Exact 3-Satisfiability (MAX-E-$3$-SAT) problem by
using deep learning methodology. This methodology allows us to create a
learning algorithm able to fix Boolean variables by using local information
obtained by the Survey Propagation algorithm. By performing an accurate
analysis, on random CNF instances of the MAX-E-$3$-SAT with several Boolean
variables, we show that this new algorithm, avoiding any decimation strategy,
can build assignments better than a random one, even if the convergence of the
messages is not found. Although this algorithm is not competitive with
state-of-the-art Maximum Satisfiability (MAX-SAT) solvers, it can solve
substantially larger and more complicated problems than it ever saw during
training.
