Yes, it is possible.

You can check this article for more information.

https://arxiv.org/pdf/physics/9705023.pdf

And this link.

http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf

Here’s an example of how a network is trained to solve Navier-Stokes equations:

https://arxiv.org/pdf/1607.03597.pdf

And video:

https://youtu.be/w71zxkniJfo

Dear Ali, I have found interesting the following paper (RG available) using Neural Network for PDE boundary problem:

Article Neural-network methods for boundary value problems with irre...

Gianluca

Thanks a lot, Dr. Argentini, Dr. Bodin and Dr. Zandsalimy, all the papers you have introduced were helpful.

If we take the theory as the point of view, it means that the problem for the PDE is infinite dimensional. So, it means, that it is not always possible approximate ( uniformly for example) the PDE with an ANN. It follows according to the theory of semigroups.

Article State Variable Reconstruction From A Reduced Set Of Measurements

In my opinion, this is the best option

González-García, R., Rico-Martínez, R., & Kevrekidis, I. G. (1998). Identification of distributed parameter systems: A neural net-based approach. Computers & Chemical Engineering, 22, S965–S968. DOI:10.1016/s0098-1354(98)00191-4 

lets consider reading these examples

http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf

https://dukespace.lib.duke.edu/dspace/handle/10161/8197

Actully it is a new active area of research.

Hi

Does a LQR controller consider uncertainty?

I'm working on a convection diffusion equaition(heat equaition) in 2D and I want to design a controller that contains uncertainty. Is a LQR controller good? if not,what is the simpelest controller that contains uncertainty?