Data-driven discovery of partial differential equation models with latent variables

Area of Science:

• Fluid dynamics
• Complex systems modeling
• Data-driven science

Background:

• Spatially extended systems often contain crucial latent variables that are difficult to measure.
• This poses a significant challenge for data-driven modeling and accurate dynamic description.
• Weakly turbulent quasi-two-dimensional Kolmogorov flow serves as a relevant case study.

Purpose of the Study:

• To illustrate the use of physical constraints for overcoming limitations in modeling systems with unmeasurable latent variables.
• To develop a data-driven approach for modeling weakly turbulent Kolmogorov flow.
• To investigate the impact of measurement noise on model reconstruction.

Main Methods:

• Employing physical constraints to eliminate latent variable terms in governing partial differential equations by increasing equation order.
• Utilizing local polynomial interpolation for data processing.
• Applying sparse regression to reconstruct system dynamics from spatiotemporal data.
• Simulating experimental measurement techniques like particle image velocimetry.

Main Results:

• Successfully eliminated latent variables by increasing the order of the governing partial differential equations.
• Demonstrated the effectiveness of local polynomial interpolation and sparse regression for handling experimental-like data.
• Identified model sensitivity to measurement noise, particularly due to high-order derivatives.

Conclusions:

• Physical constraints offer a viable method to incorporate unmeasurable variables into data-driven models.
• The proposed method is effective for modeling complex fluid dynamics like Kolmogorov flow.
• Careful consideration of measurement noise is crucial when applying high-order derivative-based models.