++Very nice. I had been thinking about brushing up on my neural net skills (they're 20years rusty), and I've bookmarked this this will be a good starting point for using PDL to do so.

My one minor nitpick: the sigmoid function you chose, the "logistic function", has a derivative that's f(x) * (1-f(x)), not x * (1-x), so you should replace your nonlin() sub with

sub nonlin {
    my ( $x, $deriv ) = @_;
    my $f = 1 / ( 1 + exp( -$x ) );
    return $f * ( 1 - $f ) if defined $deriv;
    return $f;
}
[download]

... It still trains with your slope, but with my slope, it gets there faster, so 10k training loops gives better results:

Output After Training:
[
 [ 0.0007225057]
 [0.00048051061]
 [     0.999593]
 [     0.999388]
]
[download]