Forward and inverse problems of quasi-integrable Hamiltonian system using stochastic averaging method and physics-informed neural networks

Methods combining stochastic averaging and mesh-based methods (such as finite difference method, path integral method, etc.) have been widely used to solve quasi-integrable Hamiltonian systems. However, these methods are usually limited to solving transient solutions and inverse problems. Deep neural networks (DNNs) present a viable solution to these limitations. DNNs do not have the same computational complexity as mesh-based methods, which increase exponentially with the increase in dimension. This makes DNNs capable of solving problems with higher dimensions. Furthermore, the exploration of stochastic problems characterized by intricate boundary conditions is an area that still requires in-depth investigation. In this paper, we attempt to combine stochastic averaging and Physics-informed neural networks (PINNs) for predicting the response and identifying the parameters of quasi-integrable Hamiltonian systems under various boundary conditions. Firstly, by analyzing the resonance for quasi-integrable Hamiltonian systems, the averaged stochastic differential equations (SDEs) and the averaged Fokker-Planck-Kolmogorov (FPK) equation with less dimension for resonant and non-resonant cases are derived through the stochastic averaging method, respectively. The boundary conditions associated with the averaged equations are mixed boundary conditions, including reflecting boundary, absorbing boundary, or periodic boundary. Secondly, PINNs are constructed for the forward and inverse problems of the averaged FPK equations with or without periodic boundary conditions, respectively. Finally, three numerical examples are worked out and verified by the results from Monte Carlo (MC) simulation. This work provides an effective technique for the forward and inverse problems of quasi-integrable Hamiltonian systems.