Sparse learning of stochastic dynamical equations.

Area of Science:

• Complex Systems Analysis
• Dynamical Systems Theory
• Computational Physics

Background:

• Massive datasets from complex systems necessitate efficient information extraction.
• The Sparse Identification of Nonlinear Dynamics (SINDy) framework aids in identifying governing equations from simulation data.
• Stochastic dynamical systems are vital for modeling complex biophysical processes.

Purpose of the Study:

• Extend the SINDy framework to stochastic dynamical systems.
• Establish the theoretical correctness of the extended SINDy method.
• Provide practical guidance for implementing and validating the method.

Main Methods:

• Asymptotic correctness proof for stochastic SINDy in the infinite data limit.
• Development of algorithms for solving sparse regression problems in SINDy.
• Application of cross-validation for determining optimal sparsity levels.

Main Results:

• The proposed stochastic SINDy method is asymptotically correct for both original and projected variables.
• Algorithms for sparse regression were discussed and evaluated.
• Cross-validation was shown to be essential for selecting the appropriate sparsity level.

Conclusions:

• The extended SINDy framework provides a robust method for discovering governing equations in stochastic dynamical systems.
• Practical implementation benefits from careful algorithm selection and cross-validation.
• The methodology is validated on diffusion processes, demonstrating its utility in biophysical modeling.