Parameter estimation and modeling of nonlinear dynamical systems based on Runge-Kutta physics-informed neural network

To identify the nonlinear behavior of dynamical systems subjected to external excitations such as earthquakes, explosions, and impacts, a novel method is proposed for dynamical system parameter estimation and modeling based on the combination of the physics-informed neural network (PINN) and Runge-Kutta algorithm. Drawing inspiration from classical numerical integration solution rules for differential equations, a new recurrent neural network architecture is designed for modeling. PINN cells are embedded as the basic integration units of the architecture to introduce prior physics-based biases. And the motion equations of different dynamical systems can be flexibly added to this architecture as soft constraints for neural network training. The method, referred to as the Runge-Kutta physics-informed neural network (RK-PINN), differs from black-box learning, as it models the evolutionary process of nonlinear dynamical systems. The satisfactory parameter estimation capability of the RK-PINN method was demonstrated through two illustrative examples. The results indicate that embedding physics can reduce the data required for training, and the trained network can serve as a surrogate model for the dynamical system to predict future states or responses.