Solving free-surface problems for non-shallow water using boundary and initial conditions-free physics-informed neural network (bif-PINN)

Abstract

The physics-informed neural network (PINN) has emerged as an alternative approach for solving a variety of systems, including many fluid dynamics problems. However, the original PINN method suffers from the so called ‘gradient pathology’ where gradients of the residuals from the boundary and initial conditions become stiff, and thus lead to poor results.Inspired by previous works on imposing boundary conditions as hard constraints in PINN, we apply existing techniques in a way that both boundary and initial conditions are incorporated in the governing equations such that the residuals from boundary and initial conditions are no longer required in the loss function, leading to better convergence, lower memory usage and a faster training. For simplicity, our approach is referred to as bif-PINN in this work, but we emphasise that it is an application of existing techniques for solving spatio-temporal PDE system via the PINN framework. We test bif-PINN on several Lagrangian fluid dynamics problems involving either a free-surface or nontrivial vorticity. We first validate the bif-PINN approach on the small amplitude sloshing in an open container where linear solution is valid, and then we increase the amplitude to observe the nonlinear effect. Next, we test bif-PINN on the inviscid Taylor-Green vortex problem which involves nontrivial vorticity and unsteady in the Lagrangian frame where grids can be highly distorted as time increases. Lastly, the bif-PINN approach is applied to the dam break problem which involves fluid impacting on solid wall and a subsequent rebound motion.Results using PINN and bif-PINN are compared, where the original PINN method fails to produce the correct results for all problems involving a free-surface, and a clear advantage in both performance and accuracy for the bif-PINN approach is observed in all cases.