This study centered on employing the physics-informed neural networks (PINNs) approach for resolving the time-dependent Fokker-Planck-Kolmogorov (FPK) equation for the first time, culminating in the derivation of the transient probability density function. First, we derived the FPK equation for a dynamical system driven by fractional Gaussian noise (FGN). Second, a deep learning method based on PINNs was introduced for resolving the corresponding time-dependent FPK equation. Finally, two examples under two different excitation conditions were discussed to determine the effectiveness and feasibility of the PINNs algorithm. The results show that the PINNs algorithm can get the transient solution of the system under additive and multiplicative FGN. Concurrently, the Monte Carlo approach was utilized to evaluate the precision and computational efficiency of the PINNs algorithm. We found that the different comparison results are in good consistency, which proves that the PINNs algorithm is not only efficient, but also effective and interpretable.

Considering the distributed time delay and the stochastic change of environment, this research primarily studied the dynamical behavior of the gamma-distributed delay SIR model driven by Lévy noise and Markov chain. First, it is proved that there is a unique global positive solution for stochastic model. Second, by constructing appropriate stochastic Lyapunov functions that account for regime switching, we derive a sufficient condition, $R_s>1$, indicating the existence of a stationary distribution. This suggests the disease will persist over a long period. Finally, numerical simulations confirm the theoretical findings.

Accurate interval prediction of wind speed plays a vital role in ensuring the efficiency and stability of wind power generation. Due to insufficient traditional wind speed interval prediction methods for mining nonlinear features, in this paper, a novel interval prediction method was proposed by combining improved wavelet threshold and deep learning (BiTCN-BiGRU) with the nutcracker optimization algorithm (NOA). First, NOA was used to optimize the wavelet transform (WT) and BiTCN-BiGRU. Second, we applied NOA-WT to smooth the wind speed data. Then, to capture nonlinear features of time series, phase space reconstruction (PSR) was utilized to identify chaotic characteristics of the processed data. Finally, the NOA-BiTCN-BiGRU model was built to perform wind speed interval prediction. Under the same hyperparameters and network structure settings, a comparison with other deep learning methods showed that the prediction interval coverage probability (PICP) and prediction interval mean width (PIMW) of NOA-WT-BiTCN-BiGRU model achieves the best balance, with good prediction accuracy and generalization performance. This research can provide reference and guidance for nonlinear time-series interval prediction in the real world.