============== 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
• learn function operators mapping functional parameters to solutions

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

• Physics-informed neural networks (PINN) [1]_
• The Deep Ritz method [2]_
• DeepONets [3]_ and physics-informed DeepONets [4]_
• Fourier Neural Operators (FNO) [6]_ and physics-informed FNO
• Model order reduction networks (PCANN) [7]_

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 [5]_ 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] E and Yu, "The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems", 2017 .. [3] Lu, Jin and Karniadakis, “DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators”, 2020 .. [4] Wang, Wang and Perdikaris, “Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets”, 2021 .. [5] McClenny und Braga-Neto, “Self-Adaptive Physics-Informed NeuralNetworks using a Soft Attention Mechanism”, 2020 .. [6] Zong-Yi Li et al., "Fourier Neural Operator for Parametric Partial Differential Equations", 2020 .. [7] Kaushik Bhattacharya et al., "Model Reduction And Neural Networks For Parametric PDEs", 2021