A fully-comprehensive library for Physics-Informed Deep Learning under uncertainty

📌 Requirements

• python version 3.10.* (download from here)
• virtualenv version 20.14.* (download from here)

⚙️ Installation

Windows

1. Go into the directory of your project with cd project_folder_path
2. Create an empty virtual environment with py -m venv .\my_env_name
3. Enter into the virtual environment with my_env_name\scripts\activate
4. Check that the environment is empty with pip freeze; normally, it should print nothing
5. Install the required packages from the .txt file requirements.txt with pip install -r requirements.txt
6. Run again pip freeze and check that the environment is no longer empty
7. Add the environment folder to your .gitignore (in order to avoid pushing the packages on git!)

To exit from the virtual environment, use deactivate

Mac and Linux

1. Go into the directory of your project with cd project_folder_path
2. Create an empty virtual environment with virtualenv .\my_env_name
3. Enter into the virtual environment with source my_env_name\bin\activate
4. Check that the environment is empty with pip freeze; normally, it should print nothing
5. Install the required packages from the .txt file requirements.txt with pip install -r requirements.txt
6. Run again pip freeze and check that the environment is no longer empty
7. Add the environment folder to your .gitignore (in order to avoid pushing the packages on git!)

To exit from the virtual environment, use deactivate

📂 Repository Structure

• 📁 config contains .json files which encode the options and parameter choices for the test cases proposed.
• 📁 data contains the dataset for each test case. In each subfolder, you can find .npy files storing inputs, solution and parametric field values. They are stored separately, according to the category of data they belong to (such as fitting data, collocation data...).
  In particular, we have:
  • Boundary data: dom_bnd.npy, sol_bnd.npy for the coordinates and solution values of boundaries;
  • Collocation data: dom_pde.npy for the coordinates of collocation points;
  • Fitting data: dom_sol.npy, sol_train.npy and dom_par.npy, par_train.npy for coordinate and values couples for solution and/or parametric field;
  • Test data: dom_test.npy, sol_test.npy, par_test.npy for coordinate and function values of test points.
• 📁 outs contains the results for each test case. In each subfolder, you can find the folders:
  • log with loss history and summary of the experiment options and errors in .txt files
  • plot with the plots
  • thetas with network parameters
  • values with the solution computed by the network
• 📁 src contains the source code, described in detail in the section below.

💻 Source Code

• main.py is the executable script, relying on all the below modules.
• main_data.py is a script that can be run independently from the main and it generates a new data subfolder for each new test case.
• main_loader.py is a script that can be run independently from the main and it reloads a test case already present in outs.
• 📁 setup is a module containing:
  • the class to parse command line arguments (in args.py)
  • the class to set parameters (in param.py), reading them both from the configuration files and from command line
  • data_creation.py contains the class for dataset creation starting from raw data stored in the folder data
  • data_generator.py contains the class to generate raw data
  • data_loader.py defines the data loader class (now not in use)
• 📁 utility contains technical auxiliary tasks, in particular:
  • switcher.py switches to the test case under analysis among the files contained in the three folders below
  • directories.py generates directories for data generation and results storage
  • miscellaneous.py contains utility functions for all other purposes
• 📁 algorithms is a module containing classes representing the training algorithms proposed in this project:
  • ADAM: Adaptive Moment Estimation
  • HMC: Hamiltionian Monte Carlo
  • SVGD: Stein Variational Gradient Descent
  • VI: Variational Inference
    The folder includes as well the class Trainer, used to manage the pre-training and training algorithm.
• 📁 datasets contains:
  • 📁 config contains definitions of functions and domains for generating datasets
  • 📁 template contains names and definitions of input and output for generating datasets
• 📁 equations contains the differential operators library (Operators.py) and, in separate files, the definition of physical loss and dataset pre/post processing criteria for each problem studied
• 📁 networks contains classes for each part of the Bayesian Neural Network. The network built is an instance of the class BayesNN, which inherits methods and attributes from LossNN and PredNN, having the loss computation and the prediction/post-processing functionalities, respectively. In turn, the above classes inherit from CoreNN, representing a basic fully connected network).
  Network weights and biases are instances of the class Theta, which contains the overloading of some operators for an easier managements of lists of tensors.
• 📁 postprocessing is a module with:
  • the class Plotter to generate the plots and save them in the folder outs
  • the class Storage to store and load results, uncertainty quantification study, loss history and network parameters

📚 Main References

• B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data, Liu Yang, Xuhui Meng, George Em Karniadakis, Mar 2020.
• Bayesian Physics Informed Neural Networks for real-world nonlinear dynamical systems, Kevin Linka, Amelie Schäfer, Xuhui Meng, Zongren Zou, George Em Karniadakis, and Ellen Kuhl., May 2022.
• Bayesian Physics-Informed Neural Networks for Inverse Uncertainty Quantification problems in Cardiac Electrophysiology, Master Thesis at Politecnico di Milano by Daniele Ceccarelli.

💬 Authors

• Giulia Mescolini (@giuliamesc)
• Luca Sosta (@sostaluca)

💭 Tutors

• Stefano Pagani (@StefanoPagani)
• Andrea Manzoni