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FaridMoussaoui
Posts: 365
Joined: June 20th, 2008, 10:05 am

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

So, the advantages only kick in when $d \geq 4$?
Have a look here for an application up to $d = 64$: An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical finance

Cuchulainn
Posts: 58747
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 ?

So, the advantages only kick in when $d \geq 4$?
Have a look here for an application up to $d = 64$: An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical finance
And for $d \le 3$, do we FDM or meshless?

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

I am going to read Villani's book: Optimal Transport. Fortunately, as a Geneva resident, I can borrow the book from the university mathematics library.
You can find it online here

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

So, the advantages only kick in when $d \geq 4$?
Have a look here for an application up to $d = 64$: An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical finance
And for $d \le 3$, do we FDM or meshless?
wrong typing sorry !
Last edited by JohnLeM on February 11th, 2019, 5:55 pm, edited 1 time in total.

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

So, the advantages only kick in when $d \geq 4$?
We had this conversation some years before: these algorithms converges at rate $\frac{1}{N^2}$ for call/puts, ( $\frac{1}{N}$ for digitals). The overall complexity is $N^3$. In other words : to reach an error of $\epsilon$, one needs to perform $O(\frac{1}{\epsilon^{3/2}})$ operations ($O(\frac{1}{\epsilon^{3}})$ for digitals). To compare, a straight FDM method needs $O(\frac{1}{\epsilon^{D/2}})$ operations ($O(\frac{1}{\epsilon^{D}})$ for digitals): FDM methods performs better for two dimensions (D=2) (even if I think that I could fix this, specializing the algorithms), and are worse starting at dimensions three.
Last edited by JohnLeM on February 11th, 2019, 6:25 pm, edited 4 times in total.

FaridMoussaoui
Posts: 365
Joined: June 20th, 2008, 10:05 am

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

I am going to read Villani's book: Optimal Transport. Fortunately, as a Geneva resident, I can borrow the book from the university mathematics library.
You can find it online here
Thank you. I already have the book as a pdf but I spend too much time in front of my screens. I am going to read the paper book only .

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

I am studing Jean-Marc's paper (CRAS) to implement the methodology

Does the article flesh out the algorithm to aid implementation? Or is a PhD in FEM needed? I would personally have difficulty in mapping the high-level maths to an implementation. (I did meshfree but ages ago.. AFAIR it was not as good as FDM)

I would even reduce the scope even more by taking a 1-factor problem and working it out in excruciating detail,  I prefer this approach to a top-down one.
@Cuchullain, I wrote in 2008 a paper regarding the one-dimensional case. Since I never published it, I recompiled it and put it on my drive (as you could see, I already was excited by the curse of dimensionality ...). Note, the one-dimensional scheme is still interpreted there as a FDM one (indeed, FDM methods can be interpreted as meshfree methods, as are IA - DNN ones).
Last edited by JohnLeM on February 11th, 2019, 7:13 pm, edited 1 time in total.

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

So, the advantages only kick in when $d \geq 4$?
Have a look here for an application up to $d = 64$: An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical finance
Yes, these are quite old results now. For instance we tested up to D=784 : indeed, since we can integrate any Neural Networks (NN) in this PDE framework, we tested some of these NN that are publicly available for the MNIST academic problem, that is a D=784 problem.

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

I am studing Jean-Marc's paper (CRAS) to implement the methodology

Does the article flesh out the algorithm to aid implementation? Or is a PhD in FEM needed? I would personally have difficulty in mapping the high-level maths to an implementation. (I did meshfree but ages ago.. AFAIR it was not as good as FDM)

I would even reduce the scope even more by taking a 1-factor problem and working it out in excruciating detail,  I prefer this approach to a top-down one.
I implemented these methods using C++. But it is quite a challenge (> 100k lines, 12 years ...). I think that we should include these methods for instance in TensorFlow (or another framework) to give others access to it. But that's a huge work, I only have two hands and not enough ressources to hire people :/

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

It is said that the PDE is solved by a meshfree method (Fasshauer notes) but it is not explicited which one is used.

PS1: The goal of a meshfree method and quantization is to solve the curse of dimentionality.....
PS2: Have a look to Jameson or Löhner papers on meshfree methods. They are more accurate.
Well, you know, a CRAS is limited to 7 pages, thats few space...Indeed, I just adapted Meshfree methods to mathematical Finance equations: the scheme for the Kolmogorov equations is fully described there (see eq 10 and 11). I agree, the scheme to solve Fokker-Planck equations (non linear hyperbolic / parabolic equations) is not described, but one can just use S instead of $P\circ S$ in (10) and reverse time.

Cuchulainn
Posts: 58747
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 ?

The problem with many articles is that the reader cannot reproduce or check the results.And too many lacunae.

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

Well, technically the CRAS is self-contained: there are two algorithms to implement. You can reproduce both. And we can proove theoretically the errors bounds.
Last edited by JohnLeM on February 12th, 2019, 2:19 am, edited 1 time in total.

Cuchulainn
Posts: 58747
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 ?

Well, technically the CRAS is self-contained: there is two algorithms to implement. You can reproduce all of it. And we can proove theoretically the errors bounds.
OK. Hoiw many manhours would it take to implement this algorithm, days, weeks or months?

What are the two algorithms called?

JohnLeM
Topic Author
Posts: 298
Joined: September 16th, 2008, 7:15 pm

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

Yes i agree :/ we definitely should let it more toyable.
No name. I think maybe lagrangian meshfree methods or something like that but more sexy !

Cuchulainn
Posts: 58747
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 ?

I think there's a typo on page 3; you say $\nabla^2$ is the Jacobian but it is the Laplacian operator (or Hessian which become singular/negative definite).

BTW is reference [6] available yet?

I don't get what $A \circ S$ does/is: it seem to be used a number of times without a sharp definition.

// Haven't got my head yet around article [1] and how to pull-back a measure..It uses upper case # while the article uses lower case #. Are they the same #?
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