EikonalNet

Code for the paper: Sahli Costabal F, Yang Y, Perdikaris P, Hurtado DE and Kuhl E (2020) Physics-Informed Neural Networks for Cardiac Activation Mapping. Front. Phys. 8:42. doi: 10.3389/fphy.2020.00042

Dependencies:

• tensorflow v1.0
• pyDOE: pip install pydoe
• matplotlib, numpy, scipy
• Non-parametric Entropy Estimation Toolbox: https://github.com/gregversteeg/NPEET

There are two files containing the models: models_tf.py, which is a single neural network and models_para_tf.py, which implements the randomized prior functions and trains multiple neural networks in parallel.

Within the models_tf.py there is a 2D and a 3D model:

• Eikonal2DnetCV2, which has the inputs:

  • x, y: location of the collocation points to enforce the residual penalty. Each of them must be of shape N_colloc, 1.
  • x_e, y_e: location of the data points. Each of them must be of shape N_data, 1.
  • T_e: activation times data. Must be of shape N_data, 1.
  • layers: list with the number of neurons of each layer for the activation times, for example [2,20,20,1].
  • CVlayers: list with the number of neurons of each layer for the conduction velocity, for example [2,5,5,1].
  • C: maximum conduction velocity.
  • alpha: regularization coefficient for the conduction velocity.
  • alphaL2: regularization coefficient for the weights of the neural network.
  • jobs: number of cpus to use to train the model.

• Eikonal3DnetCV2, which has the inputs:

  • X: location of the collocation points to enforce the residual penalty. For a triangular mesh, this corresponds to the centroid of each triangle. Must be of shape N_colloc, 3.
  • normals: unit vectors representing the normal of the surface at the points X. For a triangular mesh, this corresponds to the normal of each triangle. Must be of shape N_colloc, 3
  • X_e: location of the data points. Each of them must be of shape N_data, 3.
  • T_e: activation times data. Must be of shape N_data, 1.
  • layers: list with the number of neurons of each layer for the activation times, for example [3,20,20,1].
  • CVlayers: list with the number of neurons of each layer for the conduction velocity, for example [3,5,5,1].
  • C: maximum conduction velocity.
  • alpha: regularization coefficient for the conduction velocity.
  • alphaL2: regularization coefficient for the weights of the neural network.
  • jobs: number of cpus to use to train the model.

Within the models_para_tf.py there is a 2D and a 3D model as well:

• Eikonal2DnetCV2RPF, which has the additional inputs than Eikonal2DnetCV2:

  • Batch: the number of neural networks to train in parallel.
  • noise_level: which is the expected standard deviation of the measurements.

• Eikonalnet3DRPF, which has the additional inputs than Eikonal3DnetCV2:

  • Batch: the number of neural networks to train in parallel.
  • noise_level: which is the expected standard deviation of the measurements.

The files 2Dexample.ipynb demonstrates the use of a single neural network and the file active_learning_2Dexample.py demontrates the use of multiple neural networks with active learning.