Robust data-driven discovery of governing physical laws with error bars.

Area of Science:

• Physics
• Applied Mathematics
• Data Science

Background:

• Discovering governing physical laws from noisy data is a significant challenge in science and engineering.
• Existing methods often struggle with noisy datasets and lack robust uncertainty quantification.

Purpose of the Study:

• To develop a robust data-driven approach for discovering ordinary differential equations (ODEs) and partial differential equations (PDEs).
• To enable accurate uncertainty quantification for discovered equations and predictions.

Main Methods:

• Utilizing dimensional analysis to select candidate terms for ODEs and PDEs.
• Employing threshold sparse Bayesian regression with automatic hyperparameter tuning via Bayesian inference.
• Quantifying uncertainties by providing error bars for discovered equations and predictions.

Main Results:

• Demonstrated effectiveness on diverse problems, including shallow water and Navier-Stokes equations.
• Showcased robustness with noisy data, outperforming traditional algorithms like least-squares and lasso.
• Achieved more accurate and robust data-driven predictions of dynamics with quantified uncertainties.

Conclusions:

• The proposed method offers a powerful and reliable tool for scientific discovery from data.
• It effectively addresses the challenge of discovering physical laws while quantifying associated uncertainties.
• The approach enhances predictive accuracy and robustness compared to classical regression techniques.