Hidden physics models: Machine learning of nonlinear partial differential equations

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-e...

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Vydané v:	Journal of computational physics Ročník 357; s. 125 - 141
Hlavní autori:	Raissi, Maziar, Karniadakis, George Em
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cambridge Elsevier Inc 15.03.2018 Elsevier Science Ltd
Predmet:	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 Small data Uncertainty quantification Fractional equations System identification Bayesian modeling Probabilistic machine learning
ISSN:	0021-9991, 1090-2716
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Popis
Shrnutí:	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.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0021-9991 1090-2716
DOI:	10.1016/j.jcp.2017.11.039