DeepXDE
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DeepXDE is a library for scientific machine learning. Use DeepXDE if you need a deep learning library that
 solves forward and inverse partial differential equations (PDEs) via physicsinformed neural network (PINN),
 solves forward and inverse integrodifferential equations (IDEs) via PINN,
 solves forward and inverse fractional partial differential equations (fPDEs) via fractional PINN (fPINN),
 approximates nonlinear operators via deep operator network (DeepONet),
 approximates functions from multifidelity data via multifidelity NN (MFNN),
 approximates functions from a dataset with/without constraints.
DeepXDE supports three tensor libraries as backends: TensorFlow 1.x (tensorflow.compat.v1
in TensorFlow 2.x), TensorFlow 2.x, and PyTorch. For how to select one, see Working with different backends.
Documentation: ReadTheDocs, SIAM Rev., Slides, Video
Papers on algorithms
 Solving PDEs and IDEs via PINN: SIAM Rev.
 Solving fPDEs via fPINN: SIAM J. Sci. Comput.
 Solving stochastic PDEs via NNarbitrary polynomial chaos (NNaPC): J. Comput. Phys.
 Solving inverse design/topology optimization via hPINN: arXiv
 Learning nonlinear operators via DeepONet: Nat. Mach. Intell., J. Comput. Phys., J. Comput. Phys.
 Learning from multifidelity data via MFNN: J. Comput. Phys., PNAS
Features
DeepXDE has implemented many algorithms as shown above and supports many features:
 complex domain geometries without tyranny mesh generation. The primitive geometries are interval, triangle, rectangle, polygon, disk, cuboid, and sphere. Other geometries can be constructed as constructive solid geometry (CSG) using three boolean operations: union, difference, and intersection.
 multiphysics, i.e., (timedependent) coupled PDEs.
 5 types of boundary conditions (BCs): Dirichlet, Neumann, Robin, periodic, and a general BC, which can be defined on an arbitrary domain or on a point set.
 different neural networks, such as (stacked/unstacked) fully connected neural network, residual neural network, and (spatiotemporal) multiscale fourier feature networks.
 6 sampling methods: uniform, pseudorandom, Latin hypercube sampling, Halton sequence, Hammersley sequence, and Sobol sequence. The training points can keep the same during training or be resampled every certain iterations.
 conveniently save the model during training, and load a trained model.
 uncertainty quantification using dropout.
 many different (weighted) losses, optimizers, learning rate schedules, metrics, etc.
 callbacks to monitor the internal states and statistics of the model during training, such as early stopping.
 enables the user code to be compact, resembling closely the mathematical formulation.
All the components of DeepXDE are loosely coupled, and thus DeepXDE is wellstructured and highly configurable. It is easy to customize DeepXDE to meet new demands.
Installation
DeepXDE requires one of the following backendspecific dependencies to be installed:
 TensorFlow 1.x: TensorFlow>=2.2.0
 TensorFlow 2.x: TensorFlow>=2.2.0 and TensorFlow Probability
 PyTorch: PyTorch
Then, you can install DeepXDE itself.
 Install the stable version with
pip
:
$ pip install deepxde
 Install the stable version with
conda
:
$ conda install c condaforge deepxde
 For developers, you should clone the folder to your local machine and put it along with your project scripts.
$ git clone https://github.com/lululxvi/deepxde.git

Other dependencies
Explore more
 Install and Setup
 Demos of forward problems
 Demos of inverse problems
 Demos of function approximation
 FAQ
 Research papers used DeepXDE
 API
Cite DeepXDE
If you use DeepXDE for academic research, you are encouraged to cite the following paper:
@article{lu2021deepxde,
author = {Lu, Lu and Meng, Xuhui and Mao, Zhiping and Karniadakis, George Em},
title = {{DeepXDE}: A deep learning library for solving differential equations},
journal = {SIAM Review},
volume = {63},
number = {1},
pages = {208228},
year = {2021},
doi = {10.1137/19M1274067}
}
Also, if you would like your paper to appear here, open an issue in the GitHub "Issues" section.
Contributing to DeepXDE
First off, thanks for taking the time to contribute!
 Reporting bugs. To report a bug, simply open an issue in the GitHub "Issues" section.
 Suggesting enhancements. To submit an enhancement suggestion for DeepXDE, including completely new features and minor improvements to existing functionality, let us know by opening an issue.
 Pull requests. If you made improvements to DeepXDE, fixed a bug, or had a new example, feel free to send us a pullrequest.
 Asking questions. To get help on how to use DeepXDE or its functionalities, you can as well open an issue.
 Answering questions. If you know the answer to any question in the "Issues", you are welcomed to answer.
The Team
DeepXDE was originally developed by Lu Lu at Brown University under the supervision of Prof. George Karniadakis from 2018 to 2020, supported by PhILMs.
DeepXDE is currently maintained by Lu Group at University of Pennsylvania with major contributions coming from several talented individuals in various forms and means. A nonexhaustive but growing list needs to mention: Shunyuan Mao, Zongren Zou.