Meshfree-based physics-informed neural networks for the unsteady Oseen equations

We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations. Firstly, based on the ideas of meshfree and small sample learning, we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh. Specifically, we optimize the neural network by minimizing the loss function to satisfy the differential operators, initial condition and boundary condition. Then, we prove the convergence of the loss function and the convergence of the neural network. In addition, the feasibility and effectiveness of the method are verified by the results of numerical experiments, and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.