Augmented physics-informed Hamiltonian networks for dynamical systems under external interactions

Neural networks have emerged as powerful tools across various disciplines, yet their application in deciphering complex physical laws remains underexplored. Addressing this, we introduce augmented physics-informed Hamiltonian networks (A-PIHNs), designed to adeptly learn the physical laws in Hamiltonian systems under complex perturbations. Utilizing Helmholtz-Hodge decomposition, A-PIHNs excel in identifying Hamiltonian dynamical systems and perturbed dynamical systems, thereby capturing the underlying physical laws more effectively than contemporary models like dissipative Hamiltonian neural networks (DHNNs). Our empirical results, spanning diverse strong perturbation scenarios, attest to A-PIHNs' superior accuracy and generalization capabilities. Furthermore, we regard the A-PIHNs model as a dynamical system. Specifically, we regard the error between the recovered dynamical systems and the true dynamical systems as a small perturbation of the proposed model, and conclude that the model can well approximate the Kolmogorov-Arnold-Moser (KAM) theory. This perspective highlights the untapped potential of neural networks for uncovering complex physical phenomena.