Time-Embedding GRBFNN: Learning Transient Response of Stochastic Dynamical Systems

Abstract This paper introduces the Time-Embedding Gaussian Radial-basis Function Neural Network (TE-GRBFNN), a novel architecture designed to learn the time-varying solutions of Fokker-Planck-Kolmogorov (FPK) equations, within the framework of Physics-informed Neural Networks (PINNs). TE-GRBFNN extends the structure of GRBFNN by integrating time-varying centers and widths of Gaussian neurons, determined by two separate time-embedding neural networks. This structural change effectively addresses the challenging unbounded boundary and normalization conditions associated with FPK equations, which have posed difficulties for other deep neural network approaches. As an initial step, we employ TE-GRBFNN to capture the response statistics of first-order nonlinear Stochastic Differential Equations (SDEs). Our results demonstrate strong agreement with Monte Carlo simulations, highlighting the efficacy of TE-GRBFNN. Once trained, the proposed TE-GRBFNN architecture can provide instantaneous predictions of FPK equation solutions at any arbitrary time and location. This feature can offer significant potential for advancing methodologies in structural optimization and control.