I have a question, maybe it seems a little strange.
If we have a non-linear problem in 2 dimensions, the same problem is linear in 3 dimensions
Can non-linear physical problems be linearized and solved by increasing one dimension and then returning to their environment?
I would say they cannot, but what do you mean by "problem"? A system of equations, a dynamical system, a partial differential equation, other stuff?
Hi
This idea came to my mind when I was working on SVM (data separation using the support vector method), whereby increasing an arbitrary dimension to the test data, it is easier to separate them. Of course, I know in a linear system, that the response to individual arbitrary inputs is equal to the response to their sum, and this definition does not change with the increase of the dimension, so it is probably not possible to do this in non-linear differential equations, but if there was a way to solve non-linear equations with This was the easiest way to solve it.
In the science of machine learning with the SVM method, the support vector in the figure below, when the data is distributed non-linearly and cannot be separated by a line, it is better to separate the data by increasing the dimension. The SVM method is the equation of this It gives us the separator line or the separator superplane. I meant that in nonlinear physics problems and nonlinear differential equations, it is possible to do something similar to this in order to simplify the method, that is, by adding a new dimension or variable so that the solution of the equation becomes easier and the window towards the solution of the problems open
Something like what Lagrange's method does to solve some equations by adding a parameter
Well, it appears that the Carleman linearization does somewhat you suggest to do: it embeds the flow of a finite-dimensional nonlinear system as an infinite-dimensional linear system. Similarly, the recent works on Koopman operators and generators construct linear and bilinear systems of higher dimension to capture nonlinear dynamics.
As far as my concern, I believe the Koopman operator could be a possible solution to your question. For this purpose, a non-linear system can be transformed into a new system that is linear (without any linearization) using the Koopman linear operator as a linear function with infinite dimension. This linear system could represent all the properties of the non-linear system, though.
This article can give you an overview of this subject
Koopman Operator Dynamical Models: Learning, Analysis and Control (arxiv.org)
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