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So yes, from a mathematician's perspective, th...

janhardo — Wed, 03 Jun 2026 22:23:56 Z

So yes, from a mathematician's perspective, this is essentially a symbolic ansatz engine written in neural-network notation. This also explains why the method ultimately produces explicit formulas such as Equation (12), rather than a trained model.

@salim-barzani The term “neur...

janhardo — Wed, 03 Jun 2026 22:21:47 Z

@salim-barzani

The term “neural network” is, to a large extent, a modern rebranding of a classical ansatz method.

The real innovation does not lie in machine learning, but rather in the choice of a large parametric function family

ϕ=
k
∑
	

w
k
	

ξ
k
	

(ψ
k
	

),

from which exact soliton, rogue-wave, or lump solutions are subsequently extracted through coefficient matching.

So yes, from a mathematician's perspective, this is essentially a symbolic ansatz engine written in neural-network notation. This also explains why the method ultimately produces explicit formulas such as Equation (12), rather than a trained model.

yea i have all stuff but your help is another thin...

salim-barzani — Wed, 03 Jun 2026 21:05:49 Z

@dharr in begin i didnt know which this method is kind of that method have it is so complecated when i was start but becuase i did the Hirota bilinear method is a type of that i will try to produce a new graph by using that iteration but i am sure if i go further some special finding parameter will come up which i can't handle it i will ask question like always if that case come up thank you Dr for your help .

i will work on it for clerify all the cases and also diagrams .

meta analysis

dharr — Wed, 03 Jun 2026 20:19:08 Z

@salim-barzani I have edited it to say that it was the [4,3,1] case. Here is how I understand what you asked for.

In your question you want to know how to solve all neural net cases. You gave

(1) a paper describing the method and solving Eq 1 for some specific neural networks [4,3,1] and [4,2,2,1] with specific functions.

(2) a worksheet t1.mw for Eq 1 the [4,3,1] case with the same activation functions as in the paper, which leads to an error.

(3) a worksheet t2-true-one.mw for Eq. 1 for the [4,2,2,1] case with the same functions as in the paper.

and the comment "t1 have problem about first case of layer and t2 dont have problem".

I explained the error in t1.mw, which for some reason you did differently from t2-true-one. So you can now complete that case. In both these cases you used a different methodology before you started the neural net part, so you arrived at different solutions from in the paper.

In response you explained you have a generator for the neural network equation that can be supplied the the type of neural network and any desired functions.

I wanted to understand whether the paper method is somehow different from your method in the sense of what "term" means in "setting the coefficients of each term to zero". So I managed to replicate the paper solution for set 1 of the [4,3,1] case. So at this point you know how to solve Eq. 1 using neural networks, getting two types of solutions (your method in t2-true-one.mw and the paper method using Hirota transformations).

So now you are giving more papers and worksheets, which I am not going to read. I think you want to somehow find all cases. But the paper says "activation function which can be interpreted arbitrarily" (any function) and you already have a generator for the different types of neural network so as far as I can see you have everything you need.

If you are asking for some huge procedure that does everything, then the problem would need to be very carefully specified. Aside from the fact that there is a lot of work doing argument checking etc., I have been reluctant to make such procedures in the past for two reasons: (1) you jump between different types of equations, methodologies, numbers of parameters etc and so the problem description is insufficiently precise, and (2) I want to provide lines of code that are simple enough that you can understand them and modify them yourself (my background as an educator I guess).

great , but you use the same as paper ...

salim-barzani — Wed, 03 Jun 2026 17:51:59 Z

@dharr this is you solved is from [4-3-1] not [4-2-1] becuae u used three function term , when always i try to solve i try to do like the author but when i learn i will do by myself , you did a great job but he used a different transformation from our i try to do like yours but i dont know the result is ot run when i apply for zero, anyway this pde is strange, i try to solve another easyer than this , i got whole idea about this method but i need maybe a week for separate the parts and my coding really is bad, there is a lot type of neural method function but in most general one they use just three kind like [4-2-1] have 66 case for solving pde with different function [4-3-1] have 286 without repetition solution and [4-2-2-1] have more than 1000 function i dont want all just each one of them i want at least one of each function and try to figure out what plot he will give me , regarding this i will shared a perfect file of which i spend more than a day on it to get the coding and undrestanding is so beutifull to undrestand this , in this file even we can do liek diagram they did in the papers and i will change the pde in case for countinuos and i will update some case about [4-2-1] which all of them are 66 but i just did 5 or more , also 1 type of [4-3-1] i hope you help me to apply for other solution kind becuase they are so important specially the one which have a different kind of plot and until now no one did it i will keep contionous on topic until i figure out all the cases how work with it , i dont want to use like the author did i want do better than him and make it generalized, please watch the table in the layer generating function and watch table carfully

[4,3,1] Solution

dharr — Wed, 03 Jun 2026 15:52:23 Z

[Edit: [4,2,1] corrected to [4,3,1]]. Here is the procedure for the [4,3,1] case. I assumed the Hirota transformation result was given correctly in the paper.

Download t1.mw