Applications of finite difference‐based physics‐informed neural networks to steady incompressible isothermal and thermal flows

Abstract

AbstractPhysics‐informed neural networks (PINNs) have aroused an upsurge in scientific machine/deep learning community, especially in research areas involving solving partial differential equations. In this work, based on the multilayer perceptron neural network and finite difference (FD) method, we apply the finite difference‐based PINNs (FD‐PINNs) to simulate the steady incompressible flows. The spatial derivatives in nonlinear governing equations are discretized by the central FD scheme, and the physical boundary conditions can be exactly satisfied in a similar way to conventional fluid flow solvers. Compared to original PINNs using automatic differentiation (AD) to discretize the governing equations, FD‐PINNs hold three‐fold potential advantages in decoupling the predictive accuracy of derivatives from the network's approximating accuracy, fulfilling the physical boundary conditions, and saving computational costs when large‐size networks or meshes are used. We first attempt to make relatively fair comparisons between AD‐PINNs, FD‐PINNs and conventional flow solvers using a benchmark flow problem—steady incompressible laminar isothermal lid‐driven cavity flow across a wide range of Reynolds numbers (Re). Our results indicate that FD‐PINNs are superior to AD‐PINNs in both predictive precision, computational efficiency and training convergence. Moreover, FD‐PINNs exhibit higher accuracy than flow solvers when insufficient mesh points are used to solve high‐Re cavity flow. Even at Re = 10,000, with only 10 internal samples, FD‐PINNs can still precisely recover the flow details on a uniform mesh of size 51 × 51. Besides, the time consumption of FD‐PINNs grows much slower than that of conventional flow solvers. We also investigate the robustness of FD‐PINNs w.r.t. the noise level of samples and the effects of pressure boundary condition imposed on solid cavity walls. Furthermore, FD‐PINNs are extensively applied to predict the incompressible thermal flows: natural convective flows in a square cavity and in a concentric annulus, where the effects of governing equations on the predictive accuracy are discovered and discussed for the first time.