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Online Weak-form Sparse Identification of Partial Differential Equations.

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This study introduces an online algorithm for identifying partial differential equations (PDEs) using the weak-form sparse identification of nonlinear dynamics (WSINDy) method. The approach efficiently processes sequential data for real-time system identification, even with time-varying coefficients.

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Area of Science:

  • Computational Mathematics
  • Applied Mathematics
  • Dynamical Systems

Background:

  • Identifying partial differential equations (PDEs) from data is crucial for modeling complex phenomena.
  • Traditional methods often require complete datasets and struggle with sequential or streaming data.
  • The sparse identification of nonlinear dynamics (SINDy) algorithm offers a powerful framework for discovering governing equations.

Purpose of the Study:

  • To develop an online algorithm for identifying PDEs from sequentially arriving data snapshots.
  • To adapt the weak-form sparse identification of nonlinear dynamics (WSINDy) algorithm for streaming identification tasks.
  • To enable real-time tracking of systems with time-varying coefficients.

Main Methods:

  • The proposed method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach.
  • It utilizes a hard-thresholding strategy (proximal operator of the L0-pseudo-norm) for efficient sparse regression on noisy, sequential data.
  • The algorithm processes solution snapshots as they arrive, enabling continuous system identification.

Main Results:

  • Successfully identified PDEs for the Kuramoto-Sivashinsky equation, nonlinear wave equation with time-varying wavespeed, and linear wave equation.
  • Demonstrated capability in one, two, and three spatial dimensions.
  • Showcased effective identification and tracking of systems with abruptly changing coefficients.

Conclusions:

  • The online WSINDy algorithm provides an efficient and robust streaming alternative for PDE identification.
  • The method is suitable for high-dimensional problems and systems with dynamic parameter variations.
  • This approach advances the field of data-driven discovery of differential equations.