Response of Gaussian white noise excited oscillators with inertia nonlinearity based on the RBFNN method

Although stochastic averaging methods have proven effective in solving the responses of nonlinear oscillators with a strong stiffness term under broadband noise excitations, these methods appear to be ineffective when dealing with oscillators that have a strong inertial nonlinearity term (also known as coordinate-dependent mass) or multiple potential wells. To address this limitation, a radial basis function neural network (RBFNN) algorithm is applied to predict the responses of oscillators with both a strong inertia nonlinearity term and multiple potential wells. The well-known Gaussian functions are chosen as radial basis functions in the model. Then, the approximate stationary probability density function (PDF) is expressed as the sum of Gaussian basis functions (GBFs) with weights. The squared error of the approximate solution for the Fokker-Plank-Kolmogorov (FPK) function is minimized using the Lagrange multiplier method to determine optimal weight coefficients. Three examples are presented to demonstrate how inertia nonlinearity terms and potential wells affect the responses. The mean square errors between Monte Carlo simulations (MCS) and RBFNN predictions are provided. The results indicate that RBFNN predictions align perfectly with those obtained from MCS.