DeepXDE

DeepXDE is a library for scientific machine learning and physics-informed learning. DeepXDE includes the following algorithms:

• physics-informed neural network (PINN)
  • solving different problems
    • solving forward/inverse ordinary/partial differential equations (ODEs/PDEs) [SIAM Rev.]
    • solving forward/inverse integro-differential equations (IDEs) [SIAM Rev.]
    • fPINN: solving forward/inverse fractional PDEs (fPDEs) [SIAM J. Sci. Comput.]
    • NN-arbitrary polynomial chaos (NN-aPC): solving forward/inverse stochastic PDEs (sPDEs) [J. Comput. Phys.]
    • PINN with hard constraints (hPINN): solving inverse design/topology optimization [SIAM J. Sci. Comput.]
  • improving PINN accuracy
    • residual-based adaptive sampling [SIAM Rev., Comput. Methods Appl. Mech. Eng.]
    • gradient-enhanced PINN (gPINN) [Comput. Methods Appl. Mech. Eng.]
    • PINN with multi-scale Fourier features [Comput. Methods Appl. Mech. Eng.]
  • Slides, Video, Video in Chinese
• (physics-informed) deep operator network (DeepONet)
  • DeepONet: learning operators [Nat. Mach. Intell.]
  • DeepONet extensions, e.g., POD-DeepONet [Comput. Methods Appl. Mech. Eng.]
  • MIONet: learning multiple-input operators [SIAM J. Sci. Comput.]
  • Fourier-DeepONet [Comput. Methods Appl. Mech. Eng.], Fourier-MIONet [arXiv]
  • physics-informed DeepONet [Sci. Adv.]
  • multifidelity DeepONet [Phys. Rev. Research]
  • DeepM&Mnet: solving multiphysics and multiscale problems [J. Comput. Phys., J. Comput. Phys.]
  • Reliable extrapolation [Comput. Methods Appl. Mech. Eng.]
• multifidelity neural network (MFNN)
  • learning from multifidelity data [J. Comput. Phys., PNAS]

DeepXDE supports five tensor libraries as backends: TensorFlow 1.x (tensorflow.compat.v1 in TensorFlow 2.x), TensorFlow 2.x, PyTorch, JAX, and PaddlePaddle. For how to select one, see Working with different backends.

Documentation: ReadTheDocs

Features

DeepXDE has implemented many algorithms as shown above and supports many features:

• enables the user code to be compact, resembling closely the mathematical formulation.
• complex domain geometries without tyranny mesh generation. The primitive geometries are interval, triangle, rectangle, polygon, disk, ellipse, star-shaped, cuboid, sphere, hypercube, and hypersphere. Other geometries can be constructed as constructive solid geometry (CSG) using three boolean operations: union, difference, and intersection. DeepXDE also supports a geometry represented by a point cloud.
• 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; and approximate distance functions for hard constraints.
• 3 automatic differentiation (AD) methods to compute derivatives: reverse mode (i.e., backpropagation), forward mode, and zero coordinate shift (ZCS).
• different neural networks: fully connected neural network (FNN), stacked FNN, residual neural network, (spatio-temporal) multi-scale Fourier feature networks, etc.
• many 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 (adaptively) every certain iterations.
• 4 function spaces: power series, Chebyshev polynomial, Gaussian random field (1D/2D).
• data-parallel training on multiple GPUs.
• different optimizers: Adam, L-BFGS, etc.
• conveniently save the model during training, and load a trained model.
• callbacks to monitor the internal states and statistics of the model during training: early stopping, etc.
• uncertainty quantification using dropout.
• float16, float32, and float64.
• many other useful features: different (weighted) losses, learning rate schedules, metrics, etc.

All the components of DeepXDE are loosely coupled, and thus DeepXDE is well-structured and highly configurable. It is easy to customize DeepXDE to meet new demands.

Installation

DeepXDE requires one of the following backend-specific dependencies to be installed:

• TensorFlow 1.x: TensorFlow>=2.7.0
• TensorFlow 2.x: TensorFlow>=2.3.0, TensorFlow Probability>=0.11.0
• PyTorch: PyTorch>=2.0.0
• JAX: JAX, Flax, Optax
• PaddlePaddle: PaddlePaddle>=2.6.0

Then, you can install DeepXDE itself.

• Install the stable version with pip:

$ pip install deepxde

• Install the stable version with conda:

$ conda install -c conda-forge 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

Explore more

• Install and Setup
• Demos of function approximation
• Demos of forward problems
• Demos of inverse problems
• Demos of operator learning
• 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   = {208-228},
  year    = {2021},
  doi     = {10.1137/19M1274067}
}

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.
• 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 in the GitHub Issues.
• Pull requests. If you made improvements to DeepXDE, fixed a bug, or had a new example, feel free to send us a pull-request.
• Asking questions. To get help on how to use DeepXDE or its functionalities, you can open a discussion in the GitHub Discussions.
• Answering questions. If you know the answer to any question in the Discussions, you are welcomed to answer.

Slack. The DeepXDE Slack hosts a primary audience of moderate to experienced DeepXDE users and developers for general chat, online discussions, collaboration, etc. If you need a slack invite, please send me an email.

The Team

DeepXDE was developed by Lu Lu under the supervision of Prof. [George Karniadakis](https://www.brown.edu