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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 under
by neatze 6y ago
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.
- caleb-allen 6y agoIt seems to me that this just kicks the can down the road--from calculation to verification.
- ah- 6y agoSometimes verification is much easier than calculation.
- jfkebwjsbx 6y agoIt depends on the problem. For integration, you can just derive. For infinitely many other problems... verification is way harder.
- empath75 6y agois it usually the case that derivatives are easier to compute than integrals?
- danbruc 6y agoThe 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 https://en.wikipedia.org/wiki/Risch_algorithm
- abstrakraft 6y agoYes. 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.
- kccqzy 6y agoSymbolically, yes that's almost always the case. Numerically it's the other way around.
- 6gvONxR4sf7o 6y agoEven 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.
- AlexCoventry 6y agoCan't you just check numerical equality at a few million random points?
- currymj 6y agoThere 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.
- neatze 6y agoHow does mathematica fails; does it returns false positive or mathematica returns "no idea how to do it" ?
- currymj 6y agono 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.