WEAK SINDy: GALERKIN-BASED DATA-DRIVEN MODEL SELECTION.

Area of Science:

• Scientific computing
• Data-driven modeling
• Differential equations

Background:

• Discovering governing equations from data is crucial for scientific modeling.
• Existing methods like SINDy struggle with noisy datasets.
• Accurate equation discovery requires robust handling of measurement errors.

Purpose of the Study:

• To introduce a novel weak formulation and discretization for learning differential equations from noisy data.
• To develop a robust and accurate method for sparse recovery of governing equations.
• To improve upon the standard SINDy algorithm for noisy data scenarios.

Main Methods:

• Developed a weak formulation replacing pointwise derivative approximations with linear transformations and variance reduction.
• Introduced the weak SINDy (WSINDy) algorithm.
• Utilized integration for natural noise reduction, inspired by Schaeffer and McCalla (2017).

Main Results:

• WSINDy enables reliable model identification from data with high noise levels (ratios > 0.1).
• The algorithm reduces coefficient error, leading to accurate predictions.
• Coefficient error scales linearly with noise, ensuring high accuracy in low-noise conditions.

Conclusions:

• WSINDy offers a robust and accurate alternative to standard SINDy for noisy data.
• The method combines SINDy's efficiency with noise reduction capabilities.
• WSINDy facilitates reliable sparse recovery of differential equations from empirical measurements.