Physics aware analytics for accurate state prediction of dynamical systems

Abstract

It has been successfully demonstrated that synchronisation of physical prior, like conservation laws with a conventional neural network significantly decreases the amount of training necessary to learn the dynamics of non-linear physical systems. Recent research shows how parameterisation of Lagrangian and Hamiltonian using conventional Neural Network’s weights and biases are achieved, and then executing the Euler-Lagrangian and Hamilton’s equation of motion for prediction of future state vectors proved to be more efficient than conventional Neural Networks in predicting non-linear dynamical systems. In classical mechanics, the Lagrangian formalism predicts the trajectory by solving the second-order Euler-Lagrangian equation following all the symmetries of the system while, the Hamiltonian mechanics explicitly uses the Poisson bracket of canonical position and canonical momentum to arrive at two independent first-order equations of motion. For a physical system that follows energy conservation, we obtain the momentum taking the first derivative of the Lagrangian which supplements the energy conservation via Hamiltonian Mechanics. The dynamics of the physical system converges when the generalised coordinates transform to suitable action angle variables using Poisson bracket formalism conserving the energy. In this work, we have demonstrated the prediction by integrating Hamiltonian and Lagrangian neural network leading to faster convergence with the actual dynamics from less number of training epochs. Hence, integration of HNN(Hamiltonian Neural Network) and LNN(Lagrangian Neural Network) outperforms their autonomous implementation. We have tested our proposed model in three conservative non-linear dynamical systems and found promising results.