DeepEIT: Recovery of the conductivity in 3-dimensional medium using deep neural network approximations

DeepEIT is an open-source Python-based code to solve an inverse problem for an elliptic equation and determine the conductivity using deep neural networks. Specifically, the proposed algorithm projects this problem into a finite dimensional space by approximating both the unknown conductivity and the corresponding solution using two independent neural networks, which are jointly trained by minimizing a novel loss function.

Method

Consider the following elliptic equation with Neumann boundary condition

with . In order to solve the inverse problem, we first construct two deep neural networks, one appximates the solution and the other approximates the conductivity. Then we propose a novel loss function to replace the Neumann boundary control with interior control by introducing a predifined harmonic function G.

Finally we jointly train the nerual networks and reconstruct both the solution and the conductivity.

Result

We perform the proposed algorithm on 3-dimensional medium, and the results are shown below.

• Hyper-parameters and reconstruction errors on different noise levels

• x=100
• y=100
• z=100

Dependencies

The code has been tested on

• macOS Big Sur & Python 3.8.8 & PyTorch 1.10.0
• Ubuntu 18.04.5 & 3.8.11 & PyTorch 1.9.0+cu111

Usage

usage:
    python main.py {train, test, plot} [optional arguments]

    positional arguments:
      {train,test,plot}     train | test | plot
        train               training mode
        test                testing mode
        plot                plotting mode

    optional arguments:
      -h, --help            show this help message and exit
      --device              cpu | cuda
      --seed                random seed
      --num_channels        hidden layer width of network
      --num_blocks          number of residual blocks of network
      --acti                activation function of the network
      --dim                 dimension of space
      --xmin                lower bound of Omega
      --xmax                upper bound of Omega
      --csv_path            path of csv path to store training parameteres
      --save_path           saved path of the results

Funding Sources

This work is supported by NSFC (No.11971104, 11531005) and National Key R&D Program of China (2020YFA0713800).