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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Veröffentlicht in:Journal of computational physics Jg. 357; S. 125 - 141
Hauptverfasser: Raissi, Maziar, Karniadakis, George Em
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge Elsevier Inc 15.03.2018
Elsevier Science Ltd
Schlagworte:
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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Zusammenfassung: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.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2017.11.039