Uncovering differential equations from data with hidden variables.

Area of Science:

• Dynamical systems theory
• Machine learning
• Scientific computing

Background:

• Sparse Identification of Nonlinear Dynamics (SINDy) is a powerful method for discovering governing equations from data.
• Traditional SINDy requires all system variables to be observed, limiting its applicability.
• Learning differential equations with unobserved (latent) variables is a significant challenge in scientific modeling.

Purpose of the Study:

• To extend the SINDy algorithm to handle systems of differential equations where some variables are not directly measured.
• To develop a computationally efficient method for inferring dynamics from incomplete observational data.
• To validate the proposed method's performance on both synthetic and real-world datasets.

Main Methods:

• The extended SINDy method regresses higher-order time derivatives of observed variables against a library of functions including lower-order derivatives.
• This approach implicitly accounts for the influence of unobserved variables.
• The method was evaluated using prediction accuracy on synthetic data and a real-world temperature time series dataset.

Main Results:

• The proposed extension of SINDy successfully learns systems of differential equations even with unobserved variables.
• The method achieved high-quality short-term forecasting accuracy on both synthetic and real temperature data.
• The new approach demonstrated computational speeds orders of magnitude faster than existing methods for latent variable problems.

Conclusions:

• The developed extension of SINDy effectively addresses the challenge of learning differential equations from partially observed data.
• This method offers a significant advancement in the field of data-driven discovery of dynamical systems.
• The approach provides a computationally efficient and accurate tool for modeling complex systems with latent variables.