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I'm considering using a neural network to power my enemies in a space shooter game I'm building and I'm wondering; how do you train neural networks when there is no one definitive good set of outputs for the network?

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I think you can use primary inputs such as health, current speed, and player direction/distance for inputs and then have a shooting direction, currently shooting value and either a thrust vector or turn speed an accel/decel value. A recurrent network just determines that outputs are fed back into neurons as inputs. There are lots of distinctive types of neural nets with different behavior. Some easy ones are 

Elman networks wiki.tcl.tk/15206 and

Hopfield networks en.wikipedia.org/wiki/Hopfield_network 

You can also check out AI Dynamic game difficulty balancing for various AI techniques and references.

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