# Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

## Abstract

Hejre, we introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. Moreover, the effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves.

- Authors:

- Brown Univ., Providence, RI (United States)
- Univ. of Pennsylvania, Philadelphia, PA (United States)

- Publication Date:

- Research Org.:
- Univ. of Pennsylvania, Philadelphia, PA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1595805

- Grant/Contract Number:
- [SC0019116; N66001-15-2-4055; FA9550-17-1-0013]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- [ Journal Volume: 378; Journal Issue: C; Related Information: https://github.com/PredictiveIntelligenceLab/PINNs]; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Data-driven scientific computing; Machine learning; Predictive modeling; Runge–Kutta methods; Nonlinear dynamics

### Citation Formats

```
Raissi, Maziar, Perdikaris, Paris, and Karniadakis, George Em. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. United States: N. p., 2018.
Web. doi:10.1016/j.jcp.2018.10.045.
```

```
Raissi, Maziar, Perdikaris, Paris, & Karniadakis, George Em. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. United States. doi:10.1016/j.jcp.2018.10.045.
```

```
Raissi, Maziar, Perdikaris, Paris, and Karniadakis, George Em. Sat .
"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations". United States. doi:10.1016/j.jcp.2018.10.045. https://www.osti.gov/servlets/purl/1595805.
```

```
@article{osti_1595805,
```

title = {Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},

author = {Raissi, Maziar and Perdikaris, Paris and Karniadakis, George Em},

abstractNote = {Hejre, we introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. Moreover, the effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves.},

doi = {10.1016/j.jcp.2018.10.045},

journal = {Journal of Computational Physics},

number = [C],

volume = [378],

place = {United States},

year = {2018},

month = {11}

}

Works referencing / citing this record:

##
Prediction of aerodynamic flow fields using convolutional neural networks

journal, June 2019

- Bhatnagar, Saakaar; Afshar, Yaser; Pan, Shaowu
- Computational Mechanics, Vol. 64, Issue 2