In this work, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by the $\alpha$ -stable Lévy process. We first optimize the Kullback–Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions. We then construct the variational formula based on the stationary Fokker–Planck equation using the Lagrangian multiplier. Moreover, we apply the empirical distribution to replace the stationary density, combining it with the data information, and we present the estimator of the drift function from the perspective of the process. In the numerical experiment, we investigate the effect of the different amounts of data and different $\alpha$ values. The experimental results demonstrate that the estimation result of the drift function is related to both and that the exact drift function agrees well with the estimated result. The estimation result will be better when the amount of data increases, and the estimation result is also better when the $\alpha$ value increases.

Extensive research has been conducted on models of ordinary differential equations (ODEs), yet these deterministic models often fail to capture the intricate complexities of real-world systems adequately. Thus, many studies have proposed the integration of Markov chains into nonlinear dynamical systems to account for perturbations arising from environmental changes and random variations. Notably, the field of parameter estimation for ODEs incorporating Markov chains still needs to be explored, creating a significant research gap. Therefore, the objective of this study is to investigate a comprehensive model capable of encompassing real-life scenarios. This model combines a system of ODEs with a continuous-time Markov chain, enabling the representation of a continuous system with discrete parameter switching. We present a machine discovery framework for parameter estimation in nonlinear dynamical systems with Markovian switching, effectively addressing this research gap. By incorporating Markov chains into the model, we adeptly capture the time-varying dynamics of real-life systems influenced by environmental factors. This approach enhances the applicability and realism of the research, enabling more precise representations of dynamical systems with Markovian switching in complex scenarios.

The theory of forward–backward stochastic differential equations occupies an important position in stochastic analysis and practical applications. However, the numerical solution of forward–backward stochastic differential equations, especially for high-dimensional cases, has stagnated. The development of deep learning provides ideas for its high-dimensional solution. In this paper, our focus lies on the fully coupled forward–backward stochastic differential equation. We design a neural network structure tailored to the characteristics of the equation and develop a hybrid BiGRU model for solving it. We introduce the time dimension based on the sequence nature after discretizing the FBSDE. By considering the interactions between preceding and succeeding time steps, we construct the BiGRU hybrid model. This enables us to effectively capture both long- and short-term dependencies, thus mitigating issues such as gradient vanishing and explosion. Residual learning is introduced within the neural network at each time step; the structure of the loss function is adjusted according to the properties of the equation. The model established above can effectively solve fully coupled forward–backward stochastic differential equations, effectively avoiding the effects of dimensional catastrophe, gradient vanishing, and gradient explosion problems, with higher accuracy, stronger stability, and stronger model interpretability.

This paper proposes a novel fast online methodology for outlier detection called the exception maximization outlier detection (EMOD) algorithm, which employs probabilistic models and statistical algorithms to detect abnormal patterns from the outputs of complex systems. The EMOD is based on a two-state Gaussian mixture model and demonstrates strong performance in probability anomaly detection working on real-time raw data rather than using special prior distribution information. We confirm this using the synthetic data from two numerical cases. For the real-world data, we have detected the short-circuit pattern of the circuit system using EMOD by the current and voltage output of a three-phase inverter. The EMOD also found an abnormal period due to COVID-19 in the insured unemployment data of 53 regions in the United States from 2000 to 2024. The application of EMOD to these two real-life datasets demonstrated the effectiveness and accuracy of our algorithm.