High-Dimensional Delayed Stochastic Navier-Stokes Models Characterized by Multi-Fractional Gaussian Noise: Existence, Uniqueness, and Approximate Solutions Based on Physics-Informed Deep Neural Networks

Source abstract

In this paper, we introduce novel high-dimensional delayed stochastic Navier-Stokes models (HD-DSNSMs) characterized by multi-fractional Gaussian noise (mGn) and then establish sufficient conditions for the existence and uniqueness of solutions. Since obtaining exact solutions for these models is mostly impossible, this work aims to develop an efficient approach by applying an approximation of multi-fractional Brownian motion (mBm) and then using physics-informed deep neural networks to predict solutions for the HD-DSNSMs characterized by mGn. The deep neural network parameters are optimized by minimizing a mean-squared error loss function, which is derived from the discretization of the HD-DSNSMs characterized by mGn via the standard Euler-Maruyama scheme. Optimization is carried out with the Adam algorithm, and the model employs the activation function sin. Four test problems are conducted to show the efficiency and accuracy of the method. The results demonstrate the ability of the method to extract solution patterns even when pressure and delay increase. This approach proves adaptable and remarkably efficient in solving the discussed models, as evidenced by the research outcome. Our results illustrate that the HD-DSNSMs characterized by mGn improve the classical DSNSMs.