Sparse semi-interpretable networks for discovering nonlinear dynamics with time-varying parameters and switching structures

Data-driven equation discovery has emerged as a new focal point in nonlinear dynamics. While significant progress has been made in identifying governing equations of time-invariant dynamical systems, the existing approaches still encounter considerable obstacles when applied to scenarios with time-varying parameters or switching structures. To bridge this gap, this paper introduces a new sparse semi-interpretable network architecture. The framework integrates an interpretable neural network, called SREINet, to identify the backbone of governing equations, with a hypernetwork designed to capture time-varying dynamics and structural transitions. In addition, a periodic pruning mechanism is applied to the hypernetwork’s output to enhance structural sparsity and improve robustness against noise. By performing symbolic regression on the hypernetwork's output, our approach can extract explicit, closed-form governing equations for time-variant and switching nonlinear dynamics. We validate the method across three representative systems with varying levels of nonlinearity and dimensionality, including a Kuramoto oscillator with logarithmic time-varying coupling strength. Finally, we demonstrate that the framework can also be used for early prediction of tipping points.​