Simulation of 3D turbulent flows using a discretized generative model physics-informed neural networks

Abstract

Physics-informed neural networks (PINNs) demonstrated efficacy in approximating partial differential equations (PDEs). However, challenges arise when dealing with high-dimensional PDEs, particularly when characterized by nonlinear and chaotic behavior, such as turbulent fluid flows. We introduce a novel methodology that integrates domain discretization, a generative model in the Sobolev function space ($H^1$), and a gating mechanism to effectively simulate high dimensional problems. The effectiveness of the method, Discretized Generative Model Physics-Informed Neural Networks (DG-PINN), is validated by its application to the simulation of a time-dependent 3D turbulent channel flow governed by the incompressible Navier–Stokes equations, a less explored problem in the existing literature. Domain discretization prevents error propagation by using different neural network models in different subdomains. The absence of initial conditions (IC) in subsequent time steps presents a challenge in identifying optimal network parameters. To address this, discretized generative models are used, improving the model’s overall performance. The global solutions’ regularity is enhanced compared to previous decomposition techniques by using the $H^1$ norm of error, rather than $L^2$. The effectiveness of the DG-PINN is validated through numerical test cases and compared against baseline PINNs and traditional domain decomposition PINNs. The DG-PINN demonstrates improvement in both approximation accuracy and computational efficiency, consistently maintaining accuracy even at later time instances. Moreover, the implementation of a distributed training strategy, facilitated by domain discretization, is discussed, resulting in improved convergence rates and more optimized memory usage.