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Robust and optimal sparse regression for nonlinear PDE models.

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This study introduces a novel method for identifying spatiotemporal dynamics from noisy data using sparse regression and weak formulation. The approach significantly enhances model accuracy, outperforming existing techniques across various noise levels.

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Area of Science:

  • Applied Mathematics
  • Computational Physics
  • Data Science

Background:

  • Accurate modeling of spatiotemporal dynamics is crucial in many scientific fields.
  • Identifying governing equations from noisy data presents significant challenges.
  • Existing methods often struggle with noise and limited data resolution.

Purpose of the Study:

  • To develop a robust method for identifying nonlinear partial differential equations from noisy data.
  • To improve the accuracy and efficiency of spatiotemporal dynamics modeling.
  • To establish a theoretical framework for understanding model accuracy in relation to data properties and noise.

Main Methods:

  • Utilizing sparse regression for feature selection and model identification.
  • Employing a weak formulation approach to handle noisy and discrete data.
  • Applying the method to the 4th-order Kuramoto-Sivashinsky equation as a benchmark.

Main Results:

  • Achieved accuracy orders of magnitude better than existing techniques.
  • Demonstrated optimized performance in both low and high noise regimes.
  • Derived a scaling relation connecting model accuracy to data properties (resolution, noise) and formulation parameters.

Conclusions:

  • The proposed method offers a powerful tool for discovering complex dynamical systems from observational data.
  • The derived scaling relation provides critical insights for experimental design and data acquisition.
  • This approach advances the field of scientific machine learning and equation discovery.