Radial-basis-function neural network for solving the transient probability density function under composite stochastic noise

Systems influenced by Gaussian white noise, Poisson white noise, and periodic excitation are common in many engineering and scientific applications, making accurate response prediction essential. Poisson noise introduces impulsive jumps, while high-frequency from the periodic excitation adds complexity, highlighting the need for a new approach to address these combined challenges. In this paper, we propose a novel deep learning framework, the radial-basis-function neural network (RBF-NN), specifically designed to resolve high-frequency oscillatory solutions in the transient probability density function (PDF) of systems under combined Gaussian white noise, Poisson white noise, and periodic excitation. This method utilizes a single-layer network with uniformly distributed neurons, where the distribution matches that of the variables, and employs RBFs as activation functions. The physical information of the equation is used as a constraint to solve the corresponding forward Kolmogorov equation that governs the transient PDF. The presence of Poisson white noise results in the forward Kolmogorov equation being an integro-differential equation, making it challenging to solve. In this method, the Gauss-Legendre quadrature is used to efficiently compute the integrals. Additionally, the Monte Carlo (MC) method is employed to solve the normalization constraint, providing approximate solutions through random sampling. Several numerical examples demonstrate the efficacy of the RBF-NN, which significantly outperforms standard Physics-Informed Neural Networks (PINNs) and PINNs with periodic layer (P-PINNs) in both computational efficiency and accuracy, especially in high-frequency scenarios. RBF-NN expands the applicability of PINNs under Gaussian white noise, Poisson white noise, and high-frequency periodic excitation. Furthermore, the influence of neuron spacing and the shape parameter on the performance and accuracy of the RBF-NN method is investigated. The effects of Poisson noise intensity and frequency of periodic excitation on the three models are explored, with clear advantages of the RBF-NN method demonstrated in handling composite stochastic noise with high frequency.