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Sci Adv. 2017 Apr 26;3(4):e1602614. doi: 10.1126/sciadv.1602614. eCollection 2017 Apr.

Data-driven discovery of partial differential equations.

Author information

1
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.
2
Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA.
3
Institute for Disease Modeling, 3150 139th Avenue Southeast, Bellevue, WA 98005, USA.

Abstract

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg-de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable.

KEYWORDS:

data-driven discovery; dynamical systems; partial differential equations; sparse regression

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