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:
-
[1];
[2];
- 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)
- OSTI Identifier:
- 1595805
- Alternate Identifier(s):
- OSTI ID: 1635941
- 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. https://doi.org/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. https://doi.org/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 = {Sat Nov 03 00:00:00 EDT 2018},
month = {Sat Nov 03 00:00:00 EDT 2018}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 2842 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Human-level concept learning through probabilistic program induction
journal, December 2015
- Lake, B. M.; Salakhutdinov, R.; Tenenbaum, J. B.
- Science, Vol. 350, Issue 6266
Predicting the sequence specificities of DNA- and RNA-binding proteins by deep learning
journal, July 2015
- Alipanahi, Babak; Delong, Andrew; Weirauch, Matthew T.
- Nature Biotechnology, Vol. 33, Issue 8
Inferring solutions of differential equations using noisy multi-fidelity data
journal, April 2017
- Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
- Journal of Computational Physics, Vol. 335
Machine learning of linear differential equations using Gaussian processes
journal, November 2017
- Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em
- Journal of Computational Physics, Vol. 348
Bayesian Numerical Homogenization
journal, January 2015
- Owhadi, Houman
- Multiscale Modeling & Simulation, Vol. 13, Issue 3
Brittleness of Bayesian inference under finite information in a continuous world
journal, January 2015
- Owhadi, Houman; Scovel, Clint; Sullivan, Tim
- Electronic Journal of Statistics, Vol. 9, Issue 1
Multilayer feedforward networks are universal approximators
journal, January 1989
- Hornik, Kurt; Stinchcombe, Maxwell; White, Halbert
- Neural Networks, Vol. 2, Issue 5
Spectral and finite difference solutions of the Burgers equation
journal, January 1986
- Basdevant, C.; Deville, M.; Haldenwang, P.
- Computers & Fluids, Vol. 14, Issue 1
Data-driven discovery of partial differential equations
journal, April 2017
- Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L.
- Science Advances, Vol. 3, Issue 4
Artificial neural networks for solving ordinary and partial differential equations
journal, January 1998
- Lagaris, I. E.; Likas, A.; Fotiadis, D. I.
- IEEE Transactions on Neural Networks, Vol. 9, Issue 5
A hybrid neural network-first principles approach to process modeling
journal, October 1992
- Psichogios, Dimitris C.; Ungar, Lyle H.
- AIChE Journal, Vol. 38, Issue 10
A paradigm for data-driven predictive modeling using field inversion and machine learning
journal, January 2016
- Parish, Eric J.; Duraisamy, Karthik
- Journal of Computational Physics, Vol. 305
Reynolds averaged turbulence modelling using deep neural networks with embedded invariance
journal, October 2016
- Ling, Julia; Kurzawski, Andrew; Templeton, Jeremy
- Journal of Fluid Mechanics, Vol. 807
Neural Network Modeling for Near Wall Turbulent Flow
journal, October 2002
- Milano, Michele; Koumoutsakos, Petros
- Journal of Computational Physics, Vol. 182, Issue 1
Multifidelity Information Fusion Algorithms for High-Dimensional Systems and Massive Data sets
journal, January 2016
- Perdikaris, Paris; Venturi, Daniele; Karniadakis, George Em
- SIAM Journal on Scientific Computing, Vol. 38, Issue 4
Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty
journal, August 2015
- Ling, J.; Templeton, J.
- Physics of Fluids, Vol. 27, Issue 8
Why Does Deep and Cheap Learning Work So Well?
journal, July 2017
- Lin, Henry W.; Tegmark, Max; Rolnick, David
- Journal of Statistical Physics, Vol. 168, Issue 6
Wavelet Scattering Regression of Quantum Chemical Energies
journal, January 2017
- Hirn, Matthew; Mallat, Stéphane; Poilvert, Nicolas
- Multiscale Modeling & Simulation, Vol. 15, Issue 2
Understanding deep convolutional networks
journal, April 2016
- Mallat, Stéphane
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 374, Issue 2065
On the limited memory BFGS method for large scale optimization
journal, August 1989
- Liu, Dong C.; Nocedal, Jorge
- Mathematical Programming, Vol. 45, Issue 1-3
Large Sample Properties of Simulations Using Latin Hypercube Sampling
journal, May 1987
- Stein, Michael
- Technometrics, Vol. 29, Issue 2
When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?
journal, March 1998
- Sloan, Ian H.; Woźniakowski, Henryk
- Journal of Complexity, Vol. 14, Issue 1
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
journal, March 2016
- Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan
- Proceedings of the National Academy of Sciences, Vol. 113, Issue 15
Works referencing / citing this record:
Data‐Driven Materials Science: Status, Challenges, and Perspectives
journal, September 2019
- Himanen, Lauri; Geurts, Amber; Foster, Adam Stuart
- Advanced Science, Vol. 6, Issue 21
Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network
journal, December 2019
- Hadian Rasanan, Amir Hosein; Bajalan, Nastaran; Parand, Kourosh
- Mathematical Methods in the Applied Sciences, Vol. 43, Issue 3
A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence
journal, January 2020
- Pawar, Suraj; San, Omer; Rasheed, Adil
- Theoretical and Computational Fluid Dynamics, Vol. 34, Issue 4
General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning
journal, May 2019
- Wei, Shiyin; Jin, Xiaowei; Li, Hui
- Computational Mechanics, Vol. 64, Issue 5
Prediction of aerodynamic flow fields using convolutional neural networks
journal, June 2019
- Bhatnagar, Saakaar; Afshar, Yaser; Pan, Shaowu
- Computational Mechanics, Vol. 64, Issue 2
Neural networks catching up with finite differences in solving partial differential equations in higher dimensions
journal, January 2020
- Avrutskiy, Vsevolod I.
- Neural Computing and Applications, Vol. 32, Issue 17
Multiscale Modeling Meets Machine Learning: What Can We Learn?
journal, February 2020
- Peng, Grace C. Y.; Alber, Mark; Buganza Tepole, Adrian
- Archives of Computational Methods in Engineering
Interactive design and variation of hull shapes: pros and cons of different CAD approaches
journal, September 2019
- Vernengo, Giuliano; Villa, Diego; Gaggero, Stefano
- International Journal on Interactive Design and Manufacturing (IJIDeM), Vol. 14, Issue 1
Solving partial differential equations by a supervised learning technique, applied for the reaction–diffusion equation
journal, November 2019
- Zakeri, Behzad; Khashehchi, Morteza; Samsam, Sanaz
- SN Applied Sciences, Vol. 1, Issue 12
Massive computational acceleration by using neural networks to emulate mechanism-based biological models
journal, September 2019
- Wang, Shangying; Fan, Kai; Luo, Nan
- Nature Communications, Vol. 10, Issue 1
Integrating machine learning and multiscale modeling—perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences
journal, November 2019
- Alber, Mark; Buganza Tepole, Adrian; Cannon, William R.
- npj Digital Medicine, Vol. 2, Issue 1
Deep neural network learning of complex binary sorption equilibria from molecular simulation data
journal, January 2019
- Sun, Yangzesheng; DeJaco, Robert F.; Siepmann, J. Ilja
- Chemical Science, Vol. 10, Issue 16
A deep learning enabler for nonintrusive reduced order modeling of fluid flows
journal, August 2019
- Pawar, S.; Rahman, S. M.; Vaddireddy, H.
- Physics of Fluids, Vol. 31, Issue 8
Memory embedded non-intrusive reduced order modeling of non-ergodic flows
journal, December 2019
- Ahmed, Shady E.; Rahman, Sk. Mashfiqur; San, Omer
- Physics of Fluids, Vol. 31, Issue 12
Solving Fokker-Planck equation using deep learning
journal, January 2020
- Xu, Yong; Zhang, Hao; Li, Yongge
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 30, Issue 1
Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data
journal, January 2020
- Vaddireddy, Harsha; Rasheed, Adil; Staples, Anne E.
- Physics of Fluids, Vol. 32, Issue 1
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations
journal, October 2019
- Han, Jiequn; Ma, Chao; Ma, Zheng
- Proceedings of the National Academy of Sciences, Vol. 116, Issue 44
Neural network models for the anisotropic Reynolds stress tensor in turbulent channel flow
journal, December 2019
- Fang, Rui; Sondak, David; Protopapas, Pavlos
- Journal of Turbulence, Vol. 21, Issue 9-10
Machine learning and artificial intelligence to aid climate change research and preparedness
journal, November 2019
- Huntingford, Chris; Jeffers, Elizabeth S.; Bonsall, Michael B.
- Environmental Research Letters, Vol. 14, Issue 12
Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
journal, January 2020
- Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em
- Science, Vol. 367, Issue 6481
Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network
journal, December 2019
- Mehta, Pavan Pranjivan; Pang, Guofei; Song, Fangying
- Fractional Calculus and Applied Analysis, Vol. 22, Issue 6
Data‐Driven Materials Science: Status, Challenges, and Perspectives
journal, November 2019
- Himanen, Lauri; Geurts, Amber; Foster, Adam Stuart
- Advanced Science, Vol. 7, Issue 2
Modal Analysis of Fluid Flows: Applications and Outlook
preprint, January 2019
- Taira, Kunihiko; Hemati, Maziar S.; Brunton, Steven L.
- arXiv
Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations
text, January 2019
- Han, Jiequn; Ma, Chao; Ma, Zheng
- arXiv
A deep learning enabler for non-intrusive reduced order modeling of fluid flows
text, January 2019
- Pawar, S.; Rahman, S. M.; Vaddireddy, H.
- arXiv
Data-driven materials science: status, challenges and perspectives
text, January 2019
- Himanen, Lauri; Geurts, Amber; Foster, Adam S.
- arXiv
Neural Network Models for the Anisotropic Reynolds Stress Tensor in Turbulent Channel Flow
text, January 2019
- Fang, Rui; Sondak, David; Protopapas, Pavlos
- arXiv
A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence
text, January 2019
- Pawar, Suraj; San, Omer; Rasheed, Adil
- arXiv
Memory embedded non-intrusive reduced order modeling of non-ergodic flows
text, January 2019
- Ahmed, Shady E.; Rahman, Sk. Mashfiqur; San, Omer
- arXiv
Solving Fokker-Planck equation using deep learning
text, January 2019
- Xu, Yong; Zhang, Hao; Li, Yongge
- arXiv
Multiscale modeling meets machine learning: What can we learn?
preprint, January 2019
- Peng, Grace C. Y.; Alber, Mark; Tepole, Adrian Buganza
- arXiv