Learning model description

[BijanGithub]

Dear @vavrines

In this issue, I will describe the physics we have defined. And how I will follow its training.

[BijanGithub]

Description: We want to use NN learning to extract a model from the thermal relaxation of particles inside a cavity. The group of particles with net-zero momentum are stored inside a cavity. Their collision with each other relaxes them towards the average temperature. This evolution will be traced by the NN learning models. Here is the schematics of the work:

[BijanGithub]

We already used data which was from cavities with diffusive walls. I

n order to omit the wall effect, we can also consider the specular walls which do not pose any effect on the thermal relaxation process. Hence, it can be claimed that the thermal relaxation process will be a zero-dimensional process (not dependent to spatial coordinates) and will be a temporal process. So I will continue with specular walls, in this case.

[BijanGithub]

One question that I am thinking about:

Can we look at this problem as a suitable test case for the data-driven discovery of governing equations? In other words, is there any term inside the governing equations that could be interesting to know from the NN modeling? For instance, a term related to the collision term.

[BijanGithub]

For instance, in the temporal process, the group's net velocity and displacement are zero, therefore the dependency of transient evolution of distributions can be shown to be only based on collision:

From the Boltzmann equ:

[BijanGithub]

Maybe also Kac equation is also worth to note here:

[BijanGithub]

Now the question is, Do we see any term inside the collision integral that we can extract its value, behavior, etc from the NN model? then we can claim for a data-driven discovery modeling of the collision integral.

[vavrines]

One question that I am thinking about:

Can we look at this problem as a suitable test case for the data-driven discovery of governing equations? In other words, is there any term inside the governing equations that could be interesting to know from the NN modeling? For instance, a term related to the collision term.

A choice for discovery of macroscopic constitutive relationship is done in this paper: Zhang, Jun, and Wenjun Ma. "Data-driven discovery of governing equations for fluid dynamics based on molecular simulation." Journal of Fluid Mechanics 892 (2020). Another thought would go into the context of moment equations. Maybe we can use the DSMC data to determine the number of moments required to describe the system by the Grad's approach and the extended R13, R26, etc.

[vavrines]

Now the question is, Do we see any term inside the collision integral that we can extract its value, behavior, etc from the NN model? then we can claim for a data-driven discovery modeling of the collision integral.

By doing MD study, it is feasible to determine differential cross section and collision kernel used in the Boltzmann equation. As DSMC is at the same level as Boltzmann collision operator, I'm not sure if we can achieve this goal.