Predictive Limitations of Physics-Informed Neural Networks in Vortex
  Shedding
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.