Sorry, please can you elaborate on this question

For example linear regression ( Gaussian ) is because of delays and spatial fractals ( chaos, bifurcations ) better to change with nonlinear regression and then in the threesholding take locally linear regressions ( era of big data ).

Next example, nonlinear partial differential equation, Vlasov Poisson Fokker Planck equation should be considered with linearization and then apply ODE finite dimensional equation and computers for solving this local problem.

Next example, Markov processes with delays is hard to consider with fractional equation in general case , because taking into account only one parameter . Better is consider fuzzy Markovian process and then solve it with deep learning.

Dear Danilo Rastovic

Linearization ( of nonlinear dynamics) transforms and simplifies the problem, where approximations of the solutions are accepted. Your examples are typical for this case

In general, this is not true. For instance, if the dynamical system is not hyperbolic,

we can do nothing with linearization, and a different approach should be used to study the solutions. Lyapunov methods are efficient in such cases.

Best regards

At non-hyperbolic systems ( Kolmogorov Arnold Moser theory etc. ) I think that with artificial neural networks at polynomial ( not exponential )growth, with deterministic and stochastic linearization, we can simulate succesive locally the system. See for example Ott Grebogi York theory about the stabilization of non-linear chaos. Might be I am not in right, but it is in each case, not more mathematics than machine learning and artificial intelligence in general case.

Normally, at Ott Grebogi York theory of control of chaos the goal is via intelligent perturbation apply linearization and the get the fixed point and stabilization.The fixed point theorem is equivalent to the minimum, maximum ( deterministic or stochastic case ) of entropy.

Of course, there is not any general method. I think that interactions between mathematics and artificial intelligence, deep learning and machine learning have a future.

I used Lyapunov functional in investigation of stability of integrated semigroups ( see my paper in Acta Mathematica et Informatica Univ. Ostraviensis, 1995 ). Cedric Villani, the winner of Fields Medal, used entropy estimation in his proof of general form of Landau damping.

I agree that the topic is exciting, and to study almost all natural phenomena using this technique provides significant interpretations, especially in the presence of strange attractors.

Cedric Villani used also Lypunov functional in his proof.

My contributions are simple. If we have time delay and spatial fractals, then we obtain Tsallis entropy and partial information. With additional sensors and actuators we must get the total information and Shannon entropy and then also exponential stabilization ( see my papers in Journal of fusion energy, 2007 and Journal of Physics & Astronomy 2016 and the other papers too ).