Brainy - An Artificial Neural Network 
Brainy is an Artificial Neural Network (ANN) using the Backpropagation algorithm. It was originally created as part of the Neural NFL project here, but was broken out into a gem to be more reusable.
Usage
From examples/sin.rb:
# Example using sin wave function
require_relative '../lib/brainy'
net = Brainy::Network.new(1, 3, 1, 0.25)
# training
4000.times do
i = rand(0..(Math::PI/2))
o = Math.sin(i)
net.train!([i], [o])
end
# testing
mse = 1000.times.map do
i = rand(0..(Math::PI/2))
o = Math.sin(i)
(o - net.evaluate([i]).first) ** 2
end.reduce(:+) / 1000
puts "your MSE: #{ mse.round(3) }" # smaller is better