Convergence of deep random vortex network method for simulating the two-dimensional incompressible flows

Physics-informed machine learning has become a popular approach for solving both forward and inverse problems involving partial differential equations. Most of the existing methods [e.g., physics-informed neural network (PINN)] require derivative calculations, which fail to effectively handle singular problems. However, whether from a theoretical or an engineering standpoint, understanding and handling singular problems is both important and challenging. A variant of PINN inspired by the important Navier–Stokes equations in fluid mechanics, referred to as the deep random vortex network (DRVN), has been proposed. DRVN formulates the flow simulation to an optimization problem, and the loss function is derived from the random vortex method. It minimizes the loss function to approximate the velocity of the flow. DRVN demonstrates promising performance in flow problems, especially one with singular kernels or initial conditions. This motivates us to explore its theoretical properties, particularly in relation to singular Navier–Stokes equations. We perform a thorough error analysis of DRVN on the two-dimensional incompressible flow equation in vorticity form on the torus, where the velocity is parameterized by the Fourier series. We establish the convergence rate and especially characterize its optimization error of stochastic gradient descent under gradient truncation. The theory characterizes the scaling of error with respect to hyperparameters in DRVN, and the simulations on two-dimensional Lamb–Oseen vortex and fractional Lamb–Oseen vortex verify the theoretical results. In simulations, we investigated the efficiency of DRVN under different parameterizations and optimizers and extended the original DRVN to handle more complex boundary conditions. The extensive simulations on comparison with PINNs, wall-bounded flow, and lid-driven cavity demonstrate the effectiveness of DRVN in fluid simulations.