DeepXDE
=======

`DeepXDE <https://github.com/lululxvi/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. <https://doi.org/10.1137/19M1274067>`_]
        - solving forward/inverse integro-differential equations (IDEs) [`SIAM Rev. <https://doi.org/10.1137/19M1274067>`_]
        - fPINN: solving forward/inverse fractional PDEs (fPDEs) [`SIAM J. Sci. Comput. <https://doi.org/10.1137/18M1229845>`_]
        - NN-arbitrary polynomial chaos (NN-aPC): solving forward/inverse stochastic PDEs (sPDEs) [`J. Comput. Phys. <https://doi.org/10.1016/j.jcp.2019.07.048>`_]
        - PINN with hard constraints (hPINN): solving inverse design/topology optimization [`SIAM J. Sci. Comput. <https://doi.org/10.1137/21M1397908>`_]
    - improving PINN accuracy
        - residual-based adaptive sampling [`SIAM Rev. <https://doi.org/10.1137/19M1274067>`_, `Comput. Methods Appl. Mech. Eng. <https://doi.org/10.1016/j.cma.2022.115671>`_]
        - gradient-enhanced PINN (gPINN) [`Comput. Methods Appl. Mech. Eng. <https://doi.org/10.1016/j.cma.2022.114823>`_]
        - PINN with multi-scale Fourier features [`Comput. Methods Appl. Mech. Eng. <https://doi.org/10.1016/j.cma.2021.113938>`_]
    - `Slides <https://github.com/lululxvi/tutorials/blob/master/20211210_pinn/pinn.pdf>`_, `Video <https://www.youtube.com/watch?v=Wfgr1pMA9fY&list=PL1e3Jic2_DwwJQ528agJYMEpA0oMaDSA9&index=13>`_, `Video in Chinese <http://tianyuan.xmu.edu.cn/cn/minicourses/637.html>`_
- (physics-informed) deep operator network (DeepONet)
    - DeepONet: learning operators [`Nat. Mach. Intell. <https://doi.org/10.1038/s42256-021-00302-5>`_]
    - DeepONet extensions, e.g., POD-DeepONet [`Comput. Methods Appl. Mech. Eng. <https://doi.org/10.1016/j.cma.2022.114778>`_]
    - MIONet: learning multiple-input operators [`SIAM J. Sci. Comput. <https://doi.org/10.1137/22M1477751>`_]
    - Fourier-DeepONet [`Comput. Methods Appl. Mech. Eng. <https://doi.org/10.1016/j.cma.2023.116300>`_], Fourier-MIONet [`arXiv <https://arxiv.org/abs/2303.04778>`_]
    - physics-informed DeepONet [`Sci. Adv. <https://doi.org/10.1126/sciadv.abi8605>`_]
    - multifidelity DeepONet [`Phys. Rev. Research <https://doi.org/10.1103/PhysRevResearch.4.023210>`_]
    - DeepM&Mnet: solving multiphysics and multiscale problems [`J. Comput. Phys. <https://doi.org/10.1016/j.jcp.2021.110296>`_, `J. Comput. Phys. <https://doi.org/10.1016/j.jcp.2021.110698>`_]
    - Reliable extrapolation [`Comput. Methods Appl. Mech. Eng. <https://doi.org/10.1016/j.cma.2023.116064>`_]
- multifidelity neural network (MFNN)
    - learning from multifidelity data [`J. Comput. Phys. <https://doi.org/10.1016/j.jcp.2019.109020>`_, `PNAS <https://doi.org/10.1073/pnas.1922210117>`_]

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 <https://deepxde.readthedocs.io/en/latest/user/installation.html#working-with-different-backends>`_.

**Documentation**: `ReadTheDocs <https://deepxde.readthedocs.io>`_

.. image:: images/pinn.png

.. image:: images/deeponet.png

.. image:: images/mfnn.png

.. image:: images/backend.png

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.

User guide
----------

.. toctree::
  :maxdepth: 2

  user/installation

.. toctree::
  :maxdepth: 1

  demos/function
  demos/pinn_forward
  demos/pinn_inverse
  demos/operator
  user/parallel
  user/faq

.. toctree::
  :maxdepth: 2

  user/research
  user/cite_deepxde
  user/team

API reference
-------------

If you are looking for information on a specific function, class or method, this part of the documentation is for you.

.. toctree::
  :maxdepth: 2
  :caption: API

  modules/deepxde
  modules/deepxde.data
  modules/deepxde.geometry
  modules/deepxde.gradients
  modules/deepxde.icbc
  modules/deepxde.nn
  modules/deepxde.nn.jax
  modules/deepxde.nn.paddle
  modules/deepxde.nn.pytorch
  modules/deepxde.nn.tensorflow
  modules/deepxde.nn.tensorflow_compat_v1
  modules/deepxde.optimizers
  modules/deepxde.optimizers.pytorch
  modules/deepxde.utils

Indices and tables
==================

* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`