How to solve nonhomogeneous-material‘s problem

[Howard-XDU-China]

How can DeepXDE be used to solve a steady-state heat conduction problem in heterogeneous materials?

Dear Dr. LuLu,

1.My problem:

Specifically, consider a square domain composed of two stacked heterogeneous materials, where the top edge has an inward heat flux of 200, the bottom edge is held at a constant temperature of 293.15 °C, and the side edges are insulated. The thermal conductivity in the upper square is 400, while in the lower square it is 1.38.

2.My issue

I found that when using a single neural network to predict the temperature distribution over the entire domain—since it is a steady-state problem where k*(Tₓₓ+Tᵧᵧ)=0—the thermal conductivity k only appears as a weighting coefficient in the loss function. As a result, due to the excessively large weight of 400, the neural network mistakenly assumes that the entire domain has a thermal conductivity of 400 (failing to capture the abrupt change in conductivity between the upper and lower regions). Consequently, the trained network is unable to learn the temperature jump at the material interface and the continuity of heat flux. Below is the finite element solution obtained using COMSOL.

3.Other

Furthermore, I have implemented an XPINN in PyTorch myself, which achieved results close to COMSOL’s solution. However, it seems that DeepXDE does not have an encapsulated XPINN implementation. Therefore, I would like to ask: how can I use DeepXDE to solve such heterogeneous material problems?

Any advice would be greatly appreciated. Thank you!

[lululxvi]

DeepXDE can solve it. You should compute k depending on x.