============== TorchPhysics

TorchPhysics is a Python library of (mesh-free) deep learning methods to solve differential equations. You can use TorchPhysics e.g. to

• solve ordinary and partial differential equations
• train a neural network to approximate solutions for different parameters
• solve inverse problems and interpolate external data

The following approaches are implemented using high-level concepts to make their usage as easy as possible:

• physics-informed neural networks (PINN) [1]_
• QRes [2]_
• the Deep Ritz method [3]_
• DeepONets [4] and Physics-Informed DeepONets [5]

We aim to also include further implementations in the future.

TorchPhysics is build upon the machine learning library PyTorch_.

.. _PyTorch: https://pytorch.org/

Features

The Goal of this library is to create a basic framework that can be used in many different applications and with different deep learning methods. To this end, TorchPhysics aims at a:

• modular and expandable structure
• easy to understand code and clean documentation
• intuitive and compact way to transfer the mathematical problem into code
• reliable and well tested code basis

Some built-in features are:

• mesh free domain generation. With pre implemented domain types: Point, Interval, Parallelogram, Circle, Triangle and Sphere
• loading external created objects, thanks to a soft dependency on Trimesh
  and Shapely
• creating complex domains with the boolean operators Union, Cut and Intersection and higher dimensional objects over the Cartesian product
• allowing interdependence of different domains, e.g. creating moving domains
• different point sampling methods for every domain: RandomUniform, Grid, Gaussian, Latin hypercube, Adaptive and some more for specific domains
• different operators to easily define a differential equation
• pre implemented fully connected neural network and easy implementation of additional model structures
• sequentially or parallel evaluation/training of different neural networks
• normalization layers and adaptive weights [6]_ to speed up the training process

• powerful and versatile training thanks to PyTorch Lightning_

  • many options for optimizers and learning rate control
  • monitoring the loss of individual conditions while training

.. _Trimesh: https://github.com/mikedh/trimesh .. Shapely: https://github.com/shapely/shapely .. PyTorch Lightning: https://www.pytorchlightning.ai/

Getting Started

To learn the functionality and usage of TorchPhysics we recommend to have a look at the following sections:

• Tutorial: Understanding the structure of TorchPhysics_
• Examples: Different applications with detailed explanations_
• Documentation_

.. _Tutorial: Understanding the structure of TorchPhysics: https://boschresearch.github.io/torchphysics/tutorial/tutorial_start.html .. _Examples: Different applications with detailed explanations: https://github.com/boschresearch/torchphysics/tree/main/examples .. _Documentation: https://boschresearch.github.io/torchphysics/index.html

Installation

TorchPhysics reqiueres the follwing dependencies to be installed:

• Python >= 3.8
• PyTorch_ >= 2.0.0
• PyTorch Lightning_ >= 2.0.0
• Numpy_ >= 1.20.2, < 2.0
• Matplotlib_ >= 3.0.0
• Scipy_ >= 1.6.3

To install TorchPhysics you can run the following code in any Python environment where pip is installed

.. code-block:: python

pip install torchphysics

Or by

.. code-block:: python

git clone https://github.com/boschresearch/torchphysics cd path_to_torchphysics_folder pip install .[all]

if you want to modify the code.

.. _Numpy: https://numpy.org/ .. _Matplotlib: https://matplotlib.org/ .. _Scipy: https://scipy.org/

About

TorchPhysics was originally developed by Nick Heilenkötter and Tom Freudenberg, as part of a seminar project at the University of Bremen, in cooperation with the Robert Bosch GmbH_. Special thanks belong to Felix Hildebrand, Uwe Iben, Daniel Christopher Kreuter and Johannes Mueller, at the Robert Bosch GmbH, for support and supervision while creating this library.

.. seminar project: http://www.math.uni-bremen.de/zetem/cms/detail.php?template=modellierungsseminar .. University of Bremen: https://www.uni-bremen.de/en/ .. _Robert Bosch GmbH: https://www.bosch.de/en/

Contribute

If you are missing a feature or detect a bug or unexpected behaviour while using this library, feel free to open an issue or a pull request in GitHub_ or contact the authors. Since we developed the code as a student project during a seminar, we cannot guarantee every feature to work properly. However, we are happy about all contributions since we aim to develop a reliable code basis and extend the library to include other approaches.

.. _GitHub: https://github.com/boschresearch/torchphysics

Cite TorchPhysics

If TorchPhysics has been helpful for your research, please cite:

.. code-block:: latex

@article{TorchPhysics, author = {Derick Nganyu Tanyu and Jianfeng Ning and Tom Freudenberg and Nick Heilenkötter and Andreas Rademacher and Uwe Iben and Peter Maass}, title = {Deep learning methods for partial differential equations and related parameter identification problems}, journal = {Inverse Problems}, doi = {10.1088/1361-6420/ace9d4}, year = {2023}, publisher = {IOP Publishing}, volume = {39}, number = {10}, pages = {103001}}

License

TorchPhysics uses an Apache License, see the LICENSE_ file.

.. _LICENSE: https://github.com/boschresearch/torchphysics/blob/main/LICENSE.txt

Bibliography

.. [1] Raissi, Perdikaris und Karniadakis, “Physics-informed neuralnetworks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”, 2019. .. [2] Bu and Karpatne, “Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEs”, 2021 .. [3] E and Yu, "The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems", 2017 .. [4] Lu, Jin and Karniadakis, “DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators”, 2020 .. [5] Wang, Wang and Perdikaris, “Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets”, 2021 .. [6] McClenny und Braga-Neto, “Self-Adaptive Physics-Informed NeuralNetworks using a Soft Attention Mechanism”, 2020