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Comment by neatze | Hacker News Reader

From article: “You need to be confident that it’s going to work all the time, and not just on some chosen problems” by mathematician Frédéric Gibou

If I understand correctly, mathematical solutions can be verified, while neural network solutions would be very hard, if not impossible to verify in reasonable time.

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There are three issues I can think he might be talking about:

1) if you get a solution, can you be sure that it's right? in this case, the neural network might spit out a wrong solution to the integral. However, this solution is easy to verify by just differentiating (which they do in the paper), so I don't think there's a problem here.

2) will the method always return a good answer? I don't think this is a requirement; Mathematica sometimes fails to integrate but people accept that.

3) Does the method work for larger, more interesting families of functions than the limited families tried in the paper? I think this is what Gibou is talking about, and seems like the strongest objection. This technique might just fail in some circumstances for reasons that are hard to anticipate.

How does mathematica fails; does it returns false positive or mathematica returns "no idea how to do it" ?

no false positives (except for rare bugs); it will usually either return some partial solution (like a simplified version of the input), or else just keep running for a very long time until you give up.

It depends on the problem.

For integration, you can just derive.

For infinitely many other problems... verification is way harder.

is it usually the case that derivatives are easier to compute than integrals?

The general algorithm for calculating integrals [1] is rather complex and I guess not suitable for humans so that calculating integrals sometimes looks much more like a black art then calculating derivatives does. On the other hand one could argue that there are algorithms for doing both and so there is no real difference.

[1] https://en.wikipedia.org/wiki/Risch_algorithm

Yes. Most of the time, the reason is that there is a general form for computing derivatives of products ((xy)' = x'y + xy') and composite functions (chain rule) while no such general rules exist for integrals.

Symbolically, yes that's almost always the case. Numerically it's the other way around.

Even for integrals, derivation isn't a silver bullet. You need to derive and also test for equality. Determining whether two expressions are the same (or enough the same, like a being the same as a^2/a except for a=0) can be really hard.

Can't you just check numerical equality at a few million random points?

It seems to me that this just kicks the can down the road--from calculation to verification.

Sometimes verification is much easier than calculation.

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