# PINNx: Physics-Informed Neural Networks for Scientific Machine Learning in JAX
[](https://github.com/chaobrain/pinnx/actions/workflows/build.yml)
[](https://pinnx.readthedocs.io/en/latest/?badge=latest)
[](https://badge.fury.io/py/pinnx)
[](https://github.com/chaobrain/pinnx/blob/master/LICENSE)
``PINNx`` is a library for scientific machine learning and physics-informed learning.
It is rewritten according to [DeepXDE](https://github.com/lululxvi/deepxde) but is enhanced by our
[Brain Dynamics Programming (BDP) ecosystem](https://ecosystem-for-brain-dynamics.readthedocs.io/).
For example, it leverages
[brainstate](https://brainstate.readthedocs.io/) for just-in-time compilation,
[brainunit](https://brainunit.readthedocs.io/) for dimensional analysis,
[braintools](https://braintools.readthedocs.io/) for checkpointing, loss functions, and other utilities.
``PINNx`` implements the following algorithms, but with the flexibility and efficiency of JAX:
- solving different PINN 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)]
- (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)]
## Installation
- Install the stable version with `pip`:
``` sh
pip install pinnx --upgrade
```
## Documentation
The official documentation is hosted on Read the Docs: [https://pinnx.readthedocs.io/](https://pinnx.readthedocs.io/)
## See also the BDP ecosystem
We are building the Brain Dynamics Programming ecosystem: https://ecosystem-for-brain-dynamics.readthedocs.io/
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"description": "# PINNx: Physics-Informed Neural Networks for Scientific Machine Learning in JAX\r\n\r\n\r\n[](https://github.com/chaobrain/pinnx/actions/workflows/build.yml)\r\n[](https://pinnx.readthedocs.io/en/latest/?badge=latest)\r\n[](https://badge.fury.io/py/pinnx)\r\n[](https://github.com/chaobrain/pinnx/blob/master/LICENSE)\r\n\r\n``PINNx`` is a library for scientific machine learning and physics-informed learning. \r\nIt is rewritten according to [DeepXDE](https://github.com/lululxvi/deepxde) but is enhanced by our \r\n[Brain Dynamics Programming (BDP) ecosystem](https://ecosystem-for-brain-dynamics.readthedocs.io/). \r\nFor example, it leverages \r\n[brainstate](https://brainstate.readthedocs.io/) for just-in-time compilation,\r\n[brainunit](https://brainunit.readthedocs.io/) for dimensional analysis, \r\n[braintools](https://braintools.readthedocs.io/) for checkpointing, loss functions, and other utilities.\r\n\r\n\r\n``PINNx`` implements the following algorithms, but with the flexibility and efficiency of JAX:\r\n\r\n- solving different PINN problems\r\n - solving forward/inverse ordinary/partial differential equations (ODEs/PDEs) [[SIAM Rev.](https://doi.org/10.1137/19M1274067)]\r\n - solving forward/inverse integro-differential equations (IDEs) [[SIAM Rev.](https://doi.org/10.1137/19M1274067)]\r\n - fPINN: solving forward/inverse fractional PDEs (fPDEs) [[SIAM J. Sci. Comput.](https://doi.org/10.1137/18M1229845)]\r\n - 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)]\r\n - PINN with hard constraints (hPINN): solving inverse design/topology optimization [[SIAM J. Sci. Comput.](https://doi.org/10.1137/21M1397908)]\r\n- improving PINN accuracy\r\n - residual-based adaptive\r\n sampling [[SIAM Rev.](https://doi.org/10.1137/19M1274067), [Comput. Methods Appl. Mech. Eng.](https://doi.org/10.1016/j.cma.2022.115671)]\r\n - gradient-enhanced PINN (gPINN) [[Comput. Methods Appl. Mech. Eng.](https://doi.org/10.1016/j.cma.2022.114823)]\r\n - PINN with multi-scale Fourier features [[Comput. Methods Appl. Mech. Eng.](https://doi.org/10.1016/j.cma.2021.113938)]\r\n- (physics-informed) deep operator network (DeepONet)\r\n - DeepONet: learning operators [[Nat. Mach. Intell.](https://doi.org/10.1038/s42256-021-00302-5)]\r\n - DeepONet extensions, e.g., POD-DeepONet [[Comput. Methods Appl. Mech. Eng.](https://doi.org/10.1016/j.cma.2022.114778)]\r\n - MIONet: learning multiple-input operators [[SIAM J. Sci. Comput.](https://doi.org/10.1137/22M1477751)]\r\n - Fourier-DeepONet [[Comput. Methods Appl. Mech. Eng.](https://doi.org/10.1016/j.cma.2023.116300)],\r\n Fourier-MIONet [[arXiv](https://arxiv.org/abs/2303.04778)]\r\n - physics-informed DeepONet [[Sci. Adv.](https://doi.org/10.1126/sciadv.abi8605)]\r\n - multifidelity DeepONet [[Phys. Rev. Research](https://doi.org/10.1103/PhysRevResearch.4.023210)]\r\n - 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)]\r\n - Reliable extrapolation [[Comput. Methods Appl. Mech. Eng.](https://doi.org/10.1016/j.cma.2023.116064)]\r\n- multifidelity neural network (MFNN)\r\n - learning from multifidelity data [[J. Comput. Phys.](https://doi.org/10.1016/j.jcp.2019.109020), [PNAS](https://doi.org/10.1073/pnas.1922210117)]\r\n\r\n## Installation\r\n\r\n- Install the stable version with `pip`:\r\n\r\n``` sh\r\npip install pinnx --upgrade\r\n```\r\n\r\n\r\n## Documentation\r\n\r\nThe official documentation is hosted on Read the Docs: [https://pinnx.readthedocs.io/](https://pinnx.readthedocs.io/)\r\n\r\n\r\n## See also the BDP ecosystem\r\n\r\nWe are building the Brain Dynamics Programming ecosystem: https://ecosystem-for-brain-dynamics.readthedocs.io/\r\n\r\n",
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