The coupled deep neural networks for coupling of the Stokes and Darcy–Forchheimer problems

Abstract

In this article, we present an efficient deep learning method called coupled deep neural networks (CDNNs) for coupling of the Stokes and Darcy-Forchheimer problems. Our method compiles the interface conditions of the coupled problems into the networks properly and can be served as an efficient alternative to the complex coupled problems. To impose energy conservation constraints, the CDNNs utilize simple fully connected layers and a custom loss function to perform the model training process as well as the physical property of the exact solution. The approach can be beneficial for the following reasons: Firstly, we sampled randomly and only input spatial coordinates without being restricted by the nature of samples. Secondly, our method is meshfree which makes it more efficient than the traditional methods. Finally, the method is parallel and can solve multiple variables independently at the same time. We present theoretical results to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. Some numerical experiments are performed and discussed to demonstrate the performance of the proposed method.

Three-dimensional single-particle tracking by interferometric scattering microscopy with a double-helix point spread function

“…As an example, we applied the new iSCAT microscopy to record the 3D trajectory of the microbead labeled to the agellum. This will allow more precise analysis of the uctuation in the motor dynamics 19,43 , and direct precise observation of the steps in motor rotation 44 .…”

PDNNs: The parallel deep neural networks for the Navier–Stokes equations coupled with heat equation

“…To obtain the approximate solution of a PDE by deep learning, a major step is to constrain the neural network to minimize the PDEs residual. Compared with the traditional grid based representation method, our neural networks adopt automatic differentiation 31 and does not involve any discretization of space, [33][34][35][36][37][38] and could break the curse of dimensionality. [39][40][41] As far as we know, there is a few theoretical work that proves the convergency of the neural network in the sample limit.…”