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

Qi Zhi He, Alexandre M. Tartakovsky

Research output: Contribution to journalArticlepeer-review

53 Scopus citations


We propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving the coupled advection-dispersion equation (ADE) and Darcy flow equation with space-dependent hydraulic conductivity (Formula presented.). In this approach, (Formula presented.), hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that K(x) is given by its values on a grid, and we use these values to train the K 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 ADEs, where its performance for various Péclet numbers (Pe) 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 large Pe. 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 of the PINN solution of the backward ADE.

Original languageEnglish (US)
Article numbere2020WR029479
JournalWater Resources Research
Issue number7
StatePublished - Jul 2021
Externally publishedYes

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© 2021. American Geophysical Union. All Rights Reserved.


  • backward advection-dispersion equations
  • data assimilation
  • forward advection-dispersion equations
  • physics-informed machine learning
  • source identification


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