Discovering approximate solution of nonlinear random vibration system by neural networks

This paper proposes an approximate analytical expression for the steady-state probability density function (PDF) of a nonlinear stochastic vibration system using neural networks. First, the PDF is expressed in exponential form based on the maximum entropy principle. By leveraging dimensional analysis, the exponential representation is rewritten as a linear combination of dimensionless clusters of system variables (e.g., excitation intensity, system states, and parameters). The approximate PDF is then derived by training two neural networks: the first learns the steady-state PDF, while the second optimizes the weight coefficients of the dimensionless clusters. Unlike existing case-by-case neural network approaches for solving the Fokker-Planck-Kolmogorov (FPK) equation, the proposed method generates a generalizable approximate expression applicable to diverse system parameters. To validate the method, we apply it to the Duffing oscillator, demonstrating close agreement with the exact solution. Further testing on a nonlinear damping system confirms high solution accuracy across a broad parameter range.