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 (*23*–*50*)