Parameter estimation and modeling of nonlinear dynamical systems based on Runge-Kutta Physics-Informed Neural Network

This repository contains codes and data for the following publication:

• Zhai, W., Tao, D. & Bao, Y. Parameter estimation and modeling of nonlinear dynamical systems based on Runge–Kutta physics-informed neural network. Nonlinear Dyn 111, 21117–21130 (2023).

Abstract

The repository contains the Python implementation using the PyTorch framework to parameter estimation and modeling of nonlinear dynamical systems based on Runge-Kutta Physics-Informed Neural Network approach. The method, referred to as the Runge–Kutta Physics-Informed Neural Network (RK-PINN) 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. The methodology is depicted in the figure below.

Contains

RK4PINN_for_Lorenz.ipynb contains Lorenz code for jupyternotebook
Lorenz_data_train_noise1.npy contains the Lorenz dataset used for training

Tips

Due to confidentiality restrictions in research collaboration, only the core code section of Lorenz example is shown here. The core code section of Lorenz example has already demonstrated the parameter estimation process of RK4PINN. Compared to the original paper, additional partial derivatives set to zero have been added to the physics loss function, making the code more robust and preventing it from falling into local optimality.

Citation

Please cite the following paper if you find the work relevant and useful in your research:

@article{zhai2023parameter,
  title={Parameter estimation and modeling of nonlinear dynamical systems based on Runge--Kutta physics-informed neural network},
  author={Zhai, Weida and Tao, Dongwang and Bao, Yuequan},
  journal={Nonlinear Dynamics},
  volume={111},
  number={22},
  pages={21117--21130},
  year={2023},
  publisher={Springer}
}