Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

Abstract

We propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled advection-dispersion and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one-and two-dimensional forward advection dispersion equations (ADEs), where its performance for various Péclet numbers (P e) is compared with the analytical and numerical solutions.We find that the PINN method is accurate with errors of less than 1% and outperforms some conventional discretization-based methods for P e larger than 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50% in the considered cases) the accuracy

Towards Optimally Weighted Physics-Informed Neural Networks in Ocean Modelling

“…In [24], the authors also chose empirically the weights to the loss functions in PINNs for study myocardial perfusion in MRI. On the other hand, in [8] the authors evaluate the different results given by a few amount of weights in the loss functions for the advection-diffusion equation.…”

“…In this work we study the behavior of the relative error (validation loss) with respect to the weight in the physical part, for Burgers, wave, and advection-diffusion equations. In [8] the authors study PINNs for advection-difussion equations, also considering different weights. On the other hand, it is very usual in the literature to find only the use of boundary and initial conditions as data, while a large amount of interior data is considered for data driven discovery of PDEs.…”