Estimating varying parameters in dynamical systems: a modular framework using switch detection, optimization and sparse regression

Abstract The estimation of static parameters in dynamical systems is well-established, but challenges remain for systems with parameters that vary over time or space. Assuming prior knowledge of the system’s dynamics, we aim to identify functions describing these varying parameters. Initially, we focus on cases where parameters are discretely switching piecewise constant functions. For this sub-class, we propose an algorithmic framework to detect discrete parameter switches and fit a piecewise constant model using optimization-based parameter estimation. Our framework is modular, allowing customization of switch detection, numerical integration and optimization steps. We demonstrate its utility through examples, including a time-varying promoter-gene expression model, a genetic toggle switch, parameter-switching manifolds, mixed fixed and switching parameters, non-uniform switching and models governed by PDEs like the heat and advection–diffusion equations. Binary segmentation is used for switch detection, and Nelder–Mead and Powell methods for optimization. In addition, we assess the framework’s robustness to measurement noise and time-series data resolution. For more complex cases, we extend the approach using dictionary-based sparse regression with trigonometric and polynomial functions to capture continuously varying parameters. We also compare our framework with two other similar methods using standard metrics under the condition of model-mismatch. This comprehensive framework offers a versatile solution for parameter estimation in varying dynamical systems.