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Hidden physics models: machine learning of nonlinear partial differential equations. (English) Zbl 1381.68248
Summary: 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.

68T05 Learning and adaptive systems in artificial intelligence
35G50 Systems of nonlinear higher-order PDEs
35Q30 Navier-Stokes equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
35R11 Fractional partial differential equations
60G15 Gaussian processes
68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text: DOI
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