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

Area of Science:

• Stochastic Systems Analysis
• Computational Physics
• Applied Mathematics

Background:

• Engineering and scientific systems often experience combined Gaussian white noise, Poisson white noise, and periodic excitation.
• Predicting system responses under these complex conditions is crucial but challenging due to impulsive jumps and high-frequency oscillations.
• Existing methods struggle to efficiently and accurately handle the combined effects of these diverse noise types.

Purpose of the Study:

• To introduce a novel deep learning framework, the radial-basis-function neural network (RBF-NN), for analyzing systems with combined stochastic and periodic excitations.
• To accurately resolve high-frequency oscillatory solutions in the transient probability density function (PDF) of such systems.
• To overcome the limitations of existing methods in handling composite stochastic noise with high-frequency components.

Main Methods:

• Developed a single-layer RBF-NN with uniformly distributed neurons and RBF activation functions.
• Utilized physical information as constraints to solve the forward Kolmogorov equation governing the transient PDF.
• Employed Gauss-Legendre quadrature for integral computation and Monte Carlo (MC) method for normalization constraints.

Main Results:

• The RBF-NN framework demonstrated significant improvements in computational efficiency and accuracy compared to standard Physics-Informed Neural Networks (PINNs) and periodic PINNs (P-PINNs).
• The method effectively handles high-frequency oscillatory solutions and composite stochastic noise.
• Investigated the influence of neuron spacing, shape parameter, Poisson noise intensity, and periodic excitation frequency on RBF-NN performance.

Conclusions:

• The proposed RBF-NN is a highly effective and efficient method for solving the forward Kolmogorov equation under complex noise conditions.
• It expands the applicability of neural network-based approaches to a wider range of stochastic systems, particularly those with high-frequency dynamics.
• RBF-NN offers a robust solution for accurate response prediction in challenging engineering and scientific applications.