Stochastic dynamics analysis of quasi-partially integrable Hamiltonian system based on NN-SAM

Abstract

Stochastic response and reliability analysis of quasi-partially integrable Hamiltonian systems are two important yet difficult problems due to high dimensionality and nonlinearity. Deep neural networks (DNNs) and the stochastic averaging method (SAM) can address the difficulty of system dimensionality in their own ways. This paper proposed a method called NN-SAM by merging physics-informed neural networks (PINNs) and SAM to solve the stochastic response and reliability of multi-dimensional quasi-partially integrable Hamiltonian systems. Firstly, by analyzing the resonance for quasi-partially integrable Hamiltonian systems, the averaged stochastic differential equations (SDEs) with less dimension for resonant and non-resonant cases through SAM are derived, respectively. Based on these averaged SDEs, the averaged Fokker–Planck–Kolmogorov (FPK) equation, the backward Kolmogorov (BK) equation and Pontryagin equation are obtained with mixed boundary conditions, including reflecting boundary, absorbing boundary or periodic boundary. Then, the PINNs are constructed for the response prediction and reliability assessment of the non-resonant case, including solving the averaged FPK equations, the BK equation and Pontryagin equation with or without periodic boundary conditions. For the resonant case, periodic layers are introduced as a hard constraint to the neural network to handle the periodic boundary conditions caused by resonance. Finally, two numerical examples are worked out and verified by the results from the Monte Carlo (MC) simulation. This work provides an effective technique for the stochastic response and reliability problems of quasi-partially integrable systems.