Koopman Open: a challenge in neural program synthesis

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Aidan Rocke

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Aug 17, 2021, 1:26:39 PM8/17/21
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Good evening AIT mailing list, 

I would like to invite members of this group to participate in a challenge at the 
intersection of computational number theory and neural program synthesis: 

1. The primary objective of this challenge in experimental mathematics is to query an information-theoretic lower-bound on the computational complexity of integer factorisation. 

2. Due to a natural correspondence between integer factorisation and unbounded Koopman operators, this problem is reducible to neural program synthesis for approximating the eigenfunctions of a suitable Koopman operator.

The challenge will officially start on 13/09/2021. More details shall be announced on 
Github around that date: https://github.com/AidanRocke/Koopman_Open

Cheers, 

Aidan Rocke

p.s. The details of this problem were initially discussed around February with an expert in Koopman operator theory that has a strong interest in number theory, but I am happy 
to discuss it with other scientists who have an interest in the average information gained from integer factorisation. 
rsa_function.pdf

Aidan Rocke

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Aug 19, 2021, 4:42:15 AM8/19/21
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Good morning AIT mailing list, 

I thought I would clarify why I am starting this challenge around mid-September, and not immediately. 

Given that the appropriate inductive biases are unknown, I am currently working on a general library for 
approximating Koopman operators using deep neural networks. The efficacy of this library may be 
benchmarked on different physical problems that have important connections with the distribution of primes. 

I believe such a library would considerably simplify participation in the challenge. 

Sincerely, 

Aidan Rocke

More information: http://www.hutter1.net/ait.htm
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Aidan Rocke

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Sep 12, 2021, 6:34:56 PM9/12/21
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Good evening AIT mailing list, 

Here is the Koopman library as promised: https://github.com/AidanRocke/Koopman

The main objective of this library is the exact and global linearisation of non-linear 
systems by approximating the eigenfunctions of Koopman operators using deep neural networks.
In the development of this library, which may take more time than expected, I plan to adhere 
to a couple principles: 

1. To figure out whether there are universal linear embeddings for certain classes of dynamical 
systems which may inform the development of new neural network architectures. (For the case 
of Hamiltonian systems, a natural constraint to add is that the operator is Unitary.) 

2. To reformulate certain turbulence theories such as Kolmogorov's K41 theory within 
Koopman Operator theory in order to test new hypotheses using this machine learning 
framework for approximating Koopman Operators. 

I suspect that a number of these Universality classes will improve our understanding
of the Riemann Hypothesis. 

Sincerely, 

Aidan 

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