2D Diffusion problem with multi-region domain in Deep Learning Toolbox

I have seen one lecture on this in mathworks website ( https://mathworks.com/videos/physics-informed-machine-learning-with-matlab-1747332177157.html ), where you have solved the 2D heat equation with a domain as a single region domain, and one other reference is (https://in.mathworks.com/help/pde/ug/solve-poisson-equation-on-unit-disk-using-pinn.html)

But what if we have a multi-region domain, with different region-specific coefficients, ICs, etc. (Please see attached problem pdf); so using FEM MATLAB PDE toolbox, we will have a solution as shown in the attached figure (please see);

Can someone please help me to develop the pdetoolbox+PINNs-based code to solve such a problem?

[Torsten]

Do you have an stl-file (e.g. from a CAD software) of your geometry to read into the PDE toolbox ? It should then be possible to apply different properties to different parts of the domain, but I didn't test this yet.

[Gobinda Debnath]

I have the code for the geometry in the pdetoolbox and i alredy solved in pdetoolbox with FEM; wanted to implement same in PINNs;

% geometry (units are of cm)
topLayer = [3 4 0 5 5 0 0 0 2 2];
midLayer = [3 4 2 3 3 2 1.5 1.5 2 2];
botLayer = [3 4 2.45 2.55 2.55 2.45 1.7 1.7 1.8 1.8];
gdm = [topLayer;midLayer;botLayer]';
[d1,bt] = decsg(gdm,'R1+R2+R3',['R1';'R2';'R3']');
figure
pdegplot(d1,"EdgeLabels","off", "FaceLabels","off")
axis off

[Torsten]

wanted to implement same in PINNs

Why ? Just for fun ? From the attached .pdf-file it seems that the model parameters needed to solve the problem are all known - so missing knowledge about their values isn't the problem, is it ?

[Gobinda Debnath]

Sir, with due respect, not for fun; I have provided two different links showing how PINNs are implemented in MATLAB to solve physics-informed problems. Now, using PDE Toolbox modeling support is necessary because, as seen in the references, the collocation points required for PINNs are essentially the same as the node points generated from PDE Toolbox mesh generation.

Moreover, we all know the importance of mesh-free methods. From MATLAB documentation, I have understood how to solve PDE/ODE problems in a single-region domain, but I am unable to extend this to a multi-region domain.

Furthermore, the attached PDF is provided for better understanding of the problem. All parameters are given, so it becomes a simpler diffusion problem. Now, we need to solve it and observe the diffusion of gold nanoparticles (GNP) over time. By solving this simple problem, I aim to understand how to handle multi-region problems using a mesh-free method (PINNs), so that in the future I can develop more advanced problems on my own.

I hope I could convince you. Kindly help, or please consider updating the Deep Learning Toolbox documentation with more examples.

[Torsten]

Kindly help, or please consider updating the Deep Learning Toolbox documentation with more examples.

I'm only a user of the MATLAB software, not a MATLAB employee. Forgive me if my answer sounded harsh.

The reason is: I'm not a fan of "artificial" approaches to solve problems where well-established methods exist. In my opinion, the main reason to propagate such methods is to produce publications - well knowing about their limitations and drawbacks.

On the other hand, I must admit that I don't have deeper knowledge about the state-of-the-art of PINN and advantages of PINN over FEM methods that could already be demonstrated for "real" applications.

Thus at least one constructive comment to your question: I'd try to use a single region and define the diffusion coefficient "c" by a user-defined function that depends on the spatial position (x,y) in the domain. This would mainly concern substituting "pdeCoeffs.c" here in the loss-function:

% Compute gradients of U and Laplacian of U.
gradU = dlgradient(sum(U,"all"),XY,EnableHigherDerivatives=true);
Laplacian = 0;
for i=1:dim
    gradU2 = dlgradient(sum(pdeCoeffs.c.*gradU(i,:),"all"), ...
                          XY,EnableHigherDerivatives=true);
    Laplacian = gradU2(i,:)+Laplacian;
end

A comparison of the PDE-solutions for the single-domain and multi-domain computation would also be helpful.