Solving the one dimensional vertical suspended sediment mixing equation with arbitrary eddy diffusivity profiles using temporal normalized physics-informed neural networks

Abstract

Analytical solutions are practical tools in ocean engineering, but their derivation is often constrained by the complexities of the real world. This underscores the necessity for alternative approaches. In this study, the potential of Physics-Informed Neural Networks (PINN) for solving the one-dimensional vertical suspended sediment mixing (settling-diffusion) equation which involves simplified and arbitrary vertical Ds profiles is explored. A new approach of temporal Normalized Physics-Informed Neural Networks (T-NPINN), which normalizes the time component is proposed, and it achieves a remarkable accuracy (Mean Square Error of 10−5 and Relative Error Loss of 10−4). T-NPINN also proves its ability to handle the challenges posed by long-duration spatiotemporal models, which is a formidable task for conventional PINN methods. In addition, the T-NPINN is free of the limitations of numerical methods, e.g., the susceptibility to inaccuracies stemming from the discretization and approximations intrinsic to their algorithms, particularly evident within intricate and dynamic oceanic environments. The demonstrated accuracy and versatility of T-NPINN make it a compelling complement to numerical techniques, effectively bridging the gap between analytical and numerical approaches and enriching the toolkit available for oceanic research and engineering.

A novel discretized physics-informed neural network model applied to the Navier–Stokes equations

“…A notable development is the introduction of automatic differentiation [1], which makes it possible to bypass numerical discretization and mesh generation, leading to a new machine learning-based technique known as PINN [2,3]. Since the introduction of the PINN, it has been widely used to approximate solution to PDE [4][5][6][7][8][9][10][11][12][13][14][15][16], particularly highly nonlinear ones with non-convex and oscillating behavior, such as Navier-Stokes equations, that pose challenges for traditional numerical discretization techniques. The problematic part of Navier-Stokes equations is because of the convective term in the equations introduces a nonlinearity due to the product of velocity components.…”