Combining data assimilation and machine learning to build data-driven models for unknown long time...

Area of Science:

• Computational Mathematics
• Dynamical Systems
• Scientific Machine Learning

Background:

• Modeling complex phenomena often involves multiscale dynamics, posing significant computational challenges.
• Inferring governing differential equations from observational data is crucial for scientific understanding and prediction.
• Existing methods struggle with the computational burden and interpretability of data-driven models for multiscale systems.

Purpose of the Study:

• To develop a robust method for discovering differential equations that capture long-term dynamics of multiscale phenomena.
• To leverage data assimilation and machine learning for efficient parameter estimation and law inference.
• To address the interpretability challenges in data-driven differential equation discovery.

Main Methods:

• A hybrid approach synergistically combining data assimilation (DA) for fast-scale parameter estimation and machine learning (ML) for slow-scale dynamics inference.
• Exploitation of time-scale separation to decouple dynamics, significantly reducing computational complexity.
• Development of an ML algorithm to learn parametric models from time series data and a strategy to ensure model interpretability.

Main Results:

• Demonstrated effectiveness and noise-robustness in reconstructing differential models from generated time series data.
• Successfully applied the method to a cardiovascular modeling test case, highlighting its practical applicability.
• Showcased a strategy to select unique, interpretable models from infinite equivalent representations.

Conclusions:

• The proposed DA-ML framework offers an efficient and interpretable approach to discovering differential equations for multiscale systems.
• The method effectively handles noisy data and provides a pathway for understanding complex scientific phenomena.
• This work opens promising avenues for applications in fields like biomedical engineering and systems biology.