Dynamical system response under Gaussian and Poisson white noises solved by deep neural network method with adaptive task decomposition and progressive learning strategy.

Abstract

The forward Kolmogorov equation corresponding to a system under the combined excitation of Gaussian and Poisson white noises is an integrodifferential equation (IDE). In our recent study, we introduced GL-PINNs, which integrates the Gauss-Legendre (GL) quadrature with the physics-informed neural networks (PINNs) framework for solving time-dependent IDEs. However, we observed that in scenarios such as insufficient learning of initial conditions or dynamical systems with strong temporal dependencies, GL-PINNs produced inaccurate solutions despite achieving low training loss values. This issue primarily stems from the GL-PINNs framework's failure to account for temporal causality. To address this limitation, we develop a deep neural network method called ATD-GLPINNs, which integrates adaptive task decomposition and progressive learning strategy. This approach decomposes the complex task along the time axis into an initial subtask and several extra tasks, which enable progressive learning through adaptive adjustment of task parameters. As an extension of the GL-PINNs, this algorithm adheres to temporal causality by prioritizing the training of early subtasks and dynamically allocating additional computational resources to subsequent ones. Numerical experiments demonstrate that our suggested method converges with significantly fewer training epochs compared to GL-PINNs, making it not only more efficient and robust, but also capable of reducing computational costs while improving prediction accuracy.