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Data-driven discovery of partial differential equations.

Samuel H Rudy1, Steven L Brunton2, Joshua L Proctor3

  • 1Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.

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Summary
This summary is machine-generated.

This study introduces a sparse regression technique to identify governing partial differential equations from spatiotemporal data. The method efficiently selects relevant terms, balances model complexity, and distinguishes between similar equations using minimal data.

Keywords:
data-driven discoverydynamical systemspartial differential equationssparse regression

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Area of Science:

  • Applied Mathematics
  • Computational Physics
  • Data Science

Background:

  • Discovering governing equations from data is crucial for understanding complex systems.
  • Traditional methods struggle with the vast search space of possible equations.

Purpose of the Study:

  • To develop a sparse regression method for discovering partial differential equations (PDEs) from spatiotemporal time-series measurements.
  • To create a computationally efficient and robust framework for automated scientific discovery.

Main Methods:

  • Utilizes sparsity-promoting regression to select relevant nonlinear and partial derivative terms.
  • Employs Pareto analysis to balance model complexity and accuracy.
  • Supports both Eulerian and Lagrangian measurement frameworks.

Main Results:

  • Successfully identified governing equations for canonical problems including Navier-Stokes, quantum harmonic oscillator, and diffusion.
  • Demonstrated robustness and computational efficiency.
  • Showcased the ability to disambiguate non-unique dynamical terms using multiple time-series datasets.

Conclusions:

  • The proposed sparse regression method offers a powerful tool for discovering physical laws in parameterized spatiotemporal systems.
  • This approach bypasses intractable first-principles derivations.
  • Enables accurate equation discovery even when dynamics are not immediately obvious.