Deficiency of equation-finding approach to data-driven modeling of dynamical systems
This highlights a critical flaw in data-driven modeling for chaotic dynamical systems, suggesting that equation-finding approaches may be misleading for physical interpretation.
The paper shows that for chaotic systems, sparse-optimization methods for discovering governing equations from imperfect data produce models sensitive to measurement procedures, yet all generate virtually identical chaotic attractors, revealing a limitation in equation-based modeling.
Finding the governing equations from data by sparse optimization has become a popular approach to deterministic modeling of dynamical systems. Considering the physical situations where the data can be imperfect due to disturbances and measurement errors, we show that for many chaotic systems, widely used sparse-optimization methods for discovering governing equations produce models that depend sensitively on the measurement procedure, yet all such models generate virtually identical chaotic attractors, leading to a striking limitation that challenges the conventional notion of equation-based modeling in complex dynamical systems. Calculating the Koopman spectra, we find that the different sets of equations agree in their large eigenvalues and the differences begin to appear when the eigenvalues are smaller than an equation-dependent threshold. The results suggest that finding the governing equations of the system and attempting to interpret them physically may lead to misleading conclusions. It would be more useful to work directly with the available data using, e.g., machine-learning methods.