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Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

This is a nice reply, JohnLeM. I have not seen any head-on comparison KS versus RKHS for 1-sample, 2-sample, independence tests etc. A cursory read of all this stuff is that the latter approach is more accurate (uniform convergence bounds, distribution-independent), robust, mathematically mature (Functional Analysis) and extendable.
The DS statistic based on the empirical distribution function $F_n$ comes across as rough-and-ready from a mathematical and numerical point of view (I am not a specialist on KS).The kernel method on the other hand (e.g. Gretton et al, equation (4)) allows us to define the MMD statistic ("MMD >> DS"). Indeed, it seems to work with low sample sizes. MMD(p,q) == 0 iff p == q. And it works on structured data.

A tongue-in-cheek remarks: "statistics don't do metrics, norms, Hilbert space, Cauchy-like sequences". I might be very wrong.

JohnLeM has done kernel methods for PDE.

katastrofa
Posts: 9327
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

“People worry that computers will get too smart and take over the world, but the real problem is that they’re too stupid and they’ve already taken over the world.”

Pedro Domingos, cited in Artificial Intelligence: A Guide for Thinking Humans by Melanie Mitchell

Cuchulainn
Posts: 62391
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I'm always in favour of having more thinking humans. Should we not train humans instead?

The above bland statement is soo banal. It's almost as bad as the waffle on negative probability.

// I have a book by Mitchell on GAs.

katastrofa
Posts: 9327
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

Machines are better and cheaper.

Cuchulainn
Posts: 62391
Joined: July 16th, 2004, 7:38 am
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### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I must say I have a bit of nostalgia for 60 bit supercomputers. I used it to simulate city heating Waterhammer (1st order pde (6), (7)) for The Hague a while back. Punch cards, real line printers, Fortran 66(IV).

It might come back again seeing that gas, wood and oil heating will be banned soon. Now what did I do with those punch cards.

https://www.win.tue.nl/analysis/reports/rana04-34.pdf

JohnLeM
Topic Author
Posts: 380
Joined: September 16th, 2008, 7:15 pm

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

I must say I have a bit of nostalgia for 60 bit supercomputers. I used it to simulate city heating Waterhammer (1st order pde (6), (7)) for The Hague a while back. Punch cards, real line printers, Fortran 66(IV).

It might come back again seeing that gas, wood and oil heating will be banned soon. Now what did I do with those punch cards.

https://www.win.tue.nl/analysis/reports/rana04-34.pdf

I want that laptop for christmas !

Cuchulainn
Posts: 62391
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Are Artificial Intelligence methods (AKA Neural Networks) for PDEs about to rediscover the wheel ?

And the crash barriers? They're extra.

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