Hidden physics models: Machine learning of nonlinear partial differential equations

• Published in: Journal of computational physics 2018-03, Vol.357, p.125-141
• Main Authors: Raissi, Maziar, Karniadakis, George Em
• Format: Article
• Language: English
• Subjects: Applications of mathematics; Artificial intelligence; Bayesian modeling; Complexity; Computational fluid dynamics; Computational physics; Data management; Fractional equations; Gaussian process; Machine learning; Mathematical models; Navier-Stokes equations; Nonlinear differential equations; Nonlinear equations; Normal distribution; Partial differential equations; Physics; Probabilistic inference; Probabilistic machine learning; Small data; System identification; Time dependence; Uncertainty quantification
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2017.11.039

While there is currently a lot of enthusiasm about “big data”, useful data is usually “small” and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from small data. In particular, we introduce hidden physics models, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier–Stokes, Schrödinger, Kuramoto–Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data.