Research ArticleAPPLIED MATHEMATICS

Data-driven discovery of partial differential equations

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Science Advances  26 Apr 2017:
Vol. 3, no. 4, e1602614
DOI: 10.1126/sciadv.1602614
  • Fig. 1 Steps in the PDE functional identification of nonlinear dynamics (PDE-FIND) algorithm, applied to infer the Navier-Stokes equations from data.

    (1a) Data are collected as snapshots of a solution to a PDE. (1b) Numerical derivatives are taken, and data are compiled into a large matrix Θ, incorporating candidate terms for the PDE. (1c) Sparse regressions are used to identify active terms in the PDE. (2a) For large data sets, sparse sampling may be used to reduce the size of the problem. (2b) Subsampling the data set is equivalent to taking a subset of rows from the linear system in Eq. 2. (2c) An identical sparse regression problem is formed but with fewer rows. (d) Active terms in ξ are synthesized into a PDE.

  • Fig. 2 Inferring the diffusion equation from a single Brownian motion.

    (A) Time series is broken into many short random walks that are used to construct histograms of the displacement. (B) Brownian motion trajectory following the diffusion equation. (C) Parameter error Embedded Image versus length of known time series. Blue symbols correspond to correct identification of the structure of the diffusion model ut = cuxx.

  • Fig. 3 Inferring nonlinearity via observing solutions at multiple amplitudes.

    (A) Example two-soliton solution to the KdV equation. (B) Applying our method to a single soliton solution determines that it solves the standard advection equation. (C) Looking at two completely separate solutions reveals nonlinearity.

  • Table 1 Summary of regression results for a wide range of canonical models of mathematical physics.

    In each example, the correct model structure is identified using PDE-FIND. The spatial and temporal sampling of the numerical simulation data used for the regression is given along with the error produced in the parameters of the model for both no noise and 1% noise. In the reaction-diffusion system, 0.5% noise is used. For Navier-Stokes and reaction-diffusion, the percent of data used in subsampling is also given. NLS, nonlinear Schrödinger; KS, Kuramoto-Sivashinsky.


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Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/4/e1602614/DC1

    Introduction

    PDE-FIND

    Examples

    Limitations

    fig. S1. Steps in the PDE functional identification of nonlinear dynamics (PDE-FIND) algorithm, applied to infer the Navier-Stokes equations from data.

    fig. S2. The numerical solution to the KdV equation plotted in space-time.

    fig. S3. The numerical solution to the Burgers’ equation plotted in space-time.

    fig. S4. The magnitude of the numerical solution to the Schrödinger’s equation plotted in space-time.

    fig. S5. The magnitude of the numerical solution to the nonlinear Schrödinger’s equation plotted in space-time.

    fig. S6. The numerical solution to the Kuramoto-Sivashinsky equation plotted in space-time.

    fig. S7. The numerical solution to the reaction-diffusion equation plotted in space-time.

    fig. S8. A single snapshot of the vorticity field is illustrated for the fluid flow past a cylinder.

    fig. S9. A single stochastic realization of Brownian motion.

    fig. S10. Five empirical distributions, illustrating the statistical spread of a particle’s expected location over time, are presented.

    fig. 11. Five empirical distributions, illustrating the statistical spread of a particle’s expected location over time, are presented.

    fig. S12. The numerical solution to the misidentified Kuramoto-Sivashinsky equation.

    fig. S13. The numerical solution to the misidentified nonlinear Schrödinger equation.

    fig. S14. Results of PDE-FIND applied to Burgers’ equation for varying levels of noise.

    table S1. Summary of regression results for a wide range of canonical models of mathematical physics.

    table S2. Summary of PDE-FIND for identifying the KdV equation.

    table S3. Summary of PDE-FIND for identifying Burgers’ equation.

    table S4. Summary of PDE-FIND for identifying the Schrödinger equation.

    table S5. Summary of PDE-FIND for identifying the nonlinear Schrödinger equation.

    table S6. Summary of PDE-FIND for identifying the Kuramoto-Sivashinsky equation.

    table S7. Summary of PDE-FIND for identifying reaction-diffusion equation.

    table S8. Summary of PDE-FIND for identifying the Navier-Stokes equation.

    table S9. Accuracy of PDE-FIND on Burgers’ equation with various grid sizes.

    References (2350)

  • Supplementary Materials

    This PDF file includes:

    • Introduction
    • PDE-FIND
    • Examples
    • Limitations
    • fig. S1. Steps in the PDE functional identification of nonlinear dynamics (PDEFIND)
      algorithm, applied to infer the Navier-Stokes equations from data.
    • fig. S2. The numerical solution to the KdV equation plotted in space-time.
    • fig. S3. The numerical solution to the Burgers’ equation plotted in space-time.
    • fig. S4. The magnitude of the numerical solution to the Schrödinger’s equation
      plotted in space-time.
    • fig. S5. The magnitude of the numerical solution to the nonlinear Schrödinger’s
      equation plotted in space-time.
    • fig. S6. The numerical solution to the Kuramoto-Sivashinsky equation plotted in
      space-time.
    • fig. S7. The numerical solution to the reaction-diffusion equation plotted in spacetime.
    • fig. S8. A single snapshot of the vorticity field is illustrated for the fluid flow past
      a cylinder.
    • fig. S9. A single stochastic realization of Brownian motion.
    • fig. S10. Five empirical distributions, illustrating the statistical spread of a
      particle’s expected location over time, are presented.
    • fig. S11. Five empirical distributions, illustrating the statistical spread of a
      particle’s expected location over time, are presented.
    • fig. S12. The numerical solution to the misidentified Kuramoto-Sivashinsky
      equation.
    • fig. S13. The numerical solution to the misidentified nonlinear Schrödinger
      equation.
    • fig. S14. Results of PDE-FIND applied to Burgers’ equation for varying levels of
      noise.
    • table S1. Summary of regression results for a wide range of canonical models of
      mathematical physics.
    • table S2. Summary of PDE-FIND for identifying the KdV equation.
    • table S3. Summary of PDE-FIND for identifying Burgers’ equation.
    • table S4. Summary of PDE-FIND for identifying the Schrödinger equation.
    • table S5. Summary of PDE-FIND for identifying the nonlinear Schrödinger
      equation.
    • table S6. Summary of PDE-FIND for identifying the Kuramoto-Sivashinsky
      equation.
    • table S7. Summary of PDE-FIND for identifying reaction-diffusion equation.
    • table S8. Summary of PDE-FIND for identifying the Navier-Stokes equation.
    • table S9. Accuracy of PDE-FIND on Burgers’ equation with various grid sizes.
    • References (23–50)

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