# Science Advances

## 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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