Re: genetic algorithms to perform 'integrations'

Caerwyn Esgydiad wrote:

I was reading about GAs a while back, in Scientific America I think it was,
and I remember seeing an article on the use of GAs to 'function fit' a curve
of data points.

This led me to wonder whether GAs have been used to perform 'integration' of
functions.  Does anyone know of any papers or sites somewhere on someone
having done this?

Yes, this would be relatively straightforward, although genetic programming would probably be more applicable than genetic algorithms since 1. the form of some antiderivatives is not immediately guessable based on the function that's being integrated and 2. many functions do not have closed-form antiderivatives, probably necessiting the ability to use iteration, etc. in order to find useful solutions.

Symbolic regression just takes a set of data points, then tries to find a curve that matches those points but iteratively calling the genetic program on the x value of each point to see what y value it predicts, and then computing a fitness based on the total error between the predicted and existing y values.

Symbolic integration could easily be done by just numerically integrating -- effectively just computing the area under the described curve -- these same points, then again passing each x value to the genetic program and comparing its prediction to the current running numeric integration value. Note that there is no unique antiderivative to any given function; instead, there are a family of them all differing by a constant, since d(f + C)/dx = df/dx, if f is a function and C is a constant. So in effect the figure you're computing would be

	int_0^x f(x)

Note that all you need here is a set of points to do this computation; you don't actually need to symbolically know the function f(x) in advance. (For test cases, of course, they're usually generated from a function directly.) Obviously, though, the more data points there are the better.

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