On the application of physics-informed neural networks in the modelling of roll waves

Abstract

In Fluid Mechanics, the transition to turbulence and the identification of stable coherent structures are important to predict flow properties. Free-surface flows can often show that kind of structure in the form of the so-called roll wave instability, identified as long waves propagating downstream at a constant velocity on the free surface. Mathematical, numerical, and experimental works have been widely reported in the literature as an attempt to predict the waves properties, but still, methods are either time costly, expensive, or inaccurate, thus remaining a matter of study by the scientific community. As an attempt to generate additional mathematical tools to explore the roll-waves problems, the present work brings an evaluation of the performance of Physics-Informed Neural Networks when used to model roll waves for a laminar flow of a Newtonian fluid. The objective was to understand if PINNs can be used successfully to model the behavior of roll waves based on a set of differential equations that describe them and a limited dataset obtained from high-resolution numerical results, and understand what are the challenges and highlights found in such an application. To that end, seven different PINNs were defined and trained using a small subset of available numerical results for a 2D transient flow of a Newtonian fluid, in favorable conditions for roll waves generation, using a set of differential equations that govern the roll waves i.e., the Shallow Water Equations. Then the PINNs were fed the whole domain points and predicted the waves interface heights and average streamwise velocities, which were compared to the reference numerical data to estimate the prediction errors. It was found that the PINNs can accurately predict roll waves properties such as frequency and wavelength. The PINNs also predicted wave heights and average velocities with some accuracy but performed poorly in the regions of wave peaks, having an overall better performance predicting flow height than velocity. The performance of the PINNs was also shown to decrease with the increase of the Froude number, an effect attributed to the mathematical hypothesis considered during the theoretical development of the system of equations considered.