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

• Published in: Journal of computational physics Vol. 378; pp. 686 - 707
• Main Authors: Raissi, M., Perdikaris, P., Karniadakis, G.E.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.02.2019; Elsevier Science Ltd
• Subjects: Artificial intelligence; Computational fluid dynamics; Computational physics; Data-driven scientific computing; Deep learning; Inverse problems; Machine learning; Mathematical models; Neural networks; Nonlinear differential equations; Nonlinear dynamics; Nonlinear equations; Partial differential equations; Predictive modeling; Quantum mechanics; Runge-Kutta method; Runge–Kutta methods; Water waves; Wave propagation; Nonlinear dynamics; Data-driven scientific computing; Predictive modeling; Runge–Kutta methods; Machine learning
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.10.045

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. 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. •We put forth a deep learning framework that enables the synergistic combination of mathematical models and data.•We introduce an effective mechanism for regularizing the training of deep neural networks in small data regimes.•The proposed methods enable scientific prediction and discovery from incomplete models and incomplete data.