Pi-Yueh Chuang

Pi-Yueh Chuang, Lorena A. Barba

Though PINNs (physics-informed neural networks) are now deemed as a complement to traditional CFD (computational fluid dynamics) solvers rather than a replacement, their ability to solve the Navier-Stokes equations without given data is still of great interest. This report presents our not-so-successful experiments of solving the Navier-Stokes equations with PINN as a replacement for traditional solvers. We aim to, with our experiments, prepare readers for the challenges they may face if they are interested in data-free PINN. In this work, we used two standard flow problems: 2D Taylor-Green vortex at Re=100 and 2D cylinder flow at Re=200. The PINN method solved the 2D Taylor-Green vortex problem with acceptable results, and we used this flow as an accuracy and performance benchmark. About 32 hours of training were required for the PINN method's accuracy to match the accuracy of a 16x16 finite-difference simulation, which took less than 20 seconds. The 2D cylinder flow, on the other hand, did not produce a physical solution. The PINN method behaved like a steady-flow solver and did not capture the vortex shedding phenomenon. By sharing our experience, we would like to emphasize that the PINN method is still a work-in-progress, especially in terms of solving flow problems without any given data. More work is needed to make PINN feasible for real-world problems in such applications.

Pi-Yueh Chuang, Lorena A. Barba

The recent surge of interest in physics-informed neural network (PINN) methods has led to a wave of studies that attest to their potential for solving partial differential equations (PDEs) and predicting the dynamics of physical systems. However, the predictive limitations of PINNs have not been thoroughly investigated. We look at the flow around a 2D cylinder and find that data-free PINNs are unable to predict vortex shedding. Data-driven PINN exhibits vortex shedding only while the training data (from a traditional CFD solver) is available, but reverts to the steady state solution when the data flow stops. We conducted dynamic mode decomposition and analyze the Koopman modes in the solutions obtained with PINNs versus a traditional fluid solver (PetIBM). The distribution of the Koopman eigenvalues on the complex plane suggests that PINN is numerically dispersive and diffusive. The PINN method reverts to the steady solution possibly as a consequence of spectral bias. This case study reaises concerns about the ability of PINNs to predict flows with instabilities, specifically vortex shedding. Our computational study supports the need for more theoretical work to analyze the numerical properties of PINN methods. The results in this paper are transparent and reproducible, with all data and code available in public repositories and persistent archives; links are provided in the paper repository at \url{https://github.com/barbagroup/jcs_paper_pinn}, and a Reproducibility Statement within the paper.