FEM encoder-decoder, a la V-cycle

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├── LICENSE
├── README.md
└── src
    ├── data_gen.py
    ├── encoder.py
    ├── fes.py
    ├── geo2d.py
    ├── initializers.py
    ├── layers.py
    ├── losses.py
    ├── metrics.py
    ├── playground.py
    └── regularizers.py

Here is a brief description of the files in the repository:

This file contains the ABC class DataGenerator which is used to generate the data for the training of the neural network. It constructs finite element matrices for different stable equations in NGSolve.

This class is an abstract base class for the data generation. So far, it requires the implementation of the following methods:

• make_operator -- use the FEM matrices to construct a linear operator, e.g., a scipy.linalg.LinearOperator.
• make_sparse_operator -- use the FEM matrices to construct a sparse matrix, e.g., a scipy.sparse.csr_matrix.
• make_solver -- use the FEM matrices to construct a direct solver. No example provided yet.
• make_sparse_solver -- use the FEM matrices to construct a sparse solver, e.g., a scipy.linalg.spilu.
• make_data -- small utility function to generate the data for the training. It uses the previous methods (members) to construct the data.
• from_smooth -- a class method to construct the data from a smooth function. It uses the previous methods (members) to construct the data.
• from_random -- a class method to construct the data from a random function. It uses the previous methods (members) to construct the data.
• get_gf -- a utility function to get the grid function with matching finite element space and dimensions as the data.

Current classes derived from DataGenerator:

• LaplaceDGen -- data generation for the Laplace equation.

We implement the encoder-decoder-like neural network in this file. There classes are derived from Keras' Model class. In particular, we have the PseudoVcycle and the PseudoMG classes. Cf. unpublished notes by P. S. Vassilevski.

This class implements the bilinear form for different equations in NGSolve. So far, we have a general convection-diffusion equation.

This file constructs different geometries. They are employed in fes.py to construct the finite element spaces.

• make_unit_square -- a unit square.
• make_l_shape -- an L-shaped domain.

This file contains the initializers for some variables in the neural network. This is just made to provide a constant matrix to one of the layers.

This file contains the layers for the neural network.

• LinearLayer -- a linear layer: no activation function, no bias.

This file contains the losses for the neural network.

• projected_l2_loss -- a projected L2 loss into the image of a linear operator. It requires using partial functions.
• l2_l1_loss -- a L2-L1 loss.

This file contains the metrics for the neural network. Metrics are used to evaluate the performance of the neural network, but they are not used in the training.

This file is used to test the different classes and functions in the repository.