Discovery of nonlinear dynamical systems using a Runge-Kutta inspired dictionary-based sparse regression approach

Area of Science:

• Dynamical systems modeling
• Machine learning
• Numerical analysis

Background:

• Discovering differential equations from time-dependent data is challenging, especially with noise and sparse sampling.
• Traditional methods often require derivative information, limiting their applicability.
• Black-box models lack interpretability and may not generalize well.

Purpose of the Study:

• To develop a robust method for discovering differential equations from noisy and sparsely sampled data.
• To create parsimonious and interpretable dynamical models.
• To extend the method for rational nonlinearities, parameter variations, and external inputs.

Main Methods:

• Integration of machine learning, dictionary learning, and numerical analysis.
• Utilizing a numerical integration framework that bypasses the need for derivative approximation.
• Employing a large dictionary of candidate nonlinear functions to identify parsimonious models.

Main Results:

• Successfully discovered diverse differential equations from noisy measurements.
• Demonstrated effectiveness on models including neural dynamics, Lorenz system, Michaelis-Menten kinetics, and Hopf normal form.
• The method is robust to corrupted and sparsely sampled data.

Conclusions:

• The developed approach offers an effective way to identify differential equations from real-world, imperfect data.
• The resulting parsimonious models enhance interpretability and generalization.
• The method's flexibility extends to complex biological networks and controlled systems.