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

QiZhi He, Alexandre M. Tartakovsky

arXiv:2012.11658v1physics.flu-dynphysics.comp-ph

TL;DR

Grid-based solutions may be unstable and require regularization, while the inverse-problem approach is computationally more expensive. The paper applies physics-informed neural networks to solve ADEs and the coupled Darcy equation. The formulation outperforms earlier PINN studies [40] [41]; backward ADE relative errors stay under 5% in most cases, and concentration measurements improve accuracy by more than 50% in the considered cases.

  • Problem

    Grid-based solutions may be unstable and require regularization, while the inverse-problem approach is computationally more expensive.

    Source: §1 · p. 2

  • Method

    The paper applies physics-informed neural networks to solve ADEs and the coupled Darcy equation.

    Source: §1 · p. 3, §6 · p. 27

  • Results

    The formulation outperforms earlier PINN studies [40] [41]; backward ADE relative errors stay under 5% in most cases, and concentration measurements improve accuracy by more than 50% in the considered cases.

    Source: §6 · p. 28

  • Takeaways & Limitations

    Available concentration measurements can be incorporated to improve the accuracy of the PINN solution for backward ADEs.

    Source: §6 · p. 28

    The SUPG comparison involves a method requiring calibration of its stability parameter for Pe.

    Source: §3.3 · p. 18

Abstract

from arXiv · show

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 P\'{e}clet 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 $Pe$ 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 of the PINN solution of the backward ADE.

1. Introduction

Advection-dispersion equations pose numerical challenges, particularly for advection-dominated and backward problems. This work applies PINNs to coupled Darcy-flow and ADE equations, including cases with variable velocity and anisotropic dispersion.

1. Introduction

  • Backward ADEs arise in source identification, but grid-based solutions may be unstable and require regularization, while inverse approaches can require repeated, computationally expensive forward solves.Source: §1 · p. 2
  • The authors use physics-informed neural networks (PINNs) to solve ADEs and the Darcy equation, training DNN approximations with equation residuals and initial and boundary conditions.Automatic differentiation evaluates space and time derivatives without numerical discretization.Source: §1 · pp. 3–4
  • The formulation uses a known conductivity field and time-dependent ADE constraints with initial and boundary conditions to solve the forward Darcy and ADE equations.The conductivity field is represented by grid values used to train its DNN.Source: §1 · pp. 2–3
  • The paper reports that PINNs can recover backward ADE solutions from terminal concentration and that available concentration measurements improve solution accuracy.Its stated novelty is applying PINNs to ADEs with velocity variations and an anisotropic dispersion tensor.Source: §1 · pp. 3–4

2. Problem formulation

The problem formulation represents concentration with a DNN and trains it against the ADE and its initial and boundary conditions. The framework also specifies the transport equation, training-point loss, and optimization procedure.

2.1. Advection-dispersion equation

  • The ADE describes concentration transport through diffusion, advection, and a source term, with initial and prescribed boundary conditions.The formulation distinguishes Dirichlet concentration and Neumann mass-flux boundaries.Source: §2.1 · p. 5
  • The formulation considers constant velocity and isotropic constant dispersion in Sections 3.1–3.3, then space-varying conductivity driving velocity in Section 3.4.The latter test assumes steady flow and dilution, making velocity independent of concentration and time.Source: §2.1 · p. 5

2.2. PINN approximation of the ADE solutions

  • A DNN approximates concentration from space-time coordinates, and substituting it into the ADE yields a residual minimized during training.For non-uniform velocity, a separate DNN approximates velocity under the stated independence assumption.Source: §2.2 · p. 6
  • The loss combines PDE, boundary-condition, and initial-condition residuals at their respective training points, so these conditions are enforced as penalties rather than hard constraints.Automatic differentiation evaluates the residual networks.Source: §2.2 · pp. 7–8
  • With a finite number of training points, selecting the relative PDE, boundary, and initial-condition loss weights is critical to solution accuracy.The paper investigates these relative weights and reports accurate ADE solutions when boundary and initial-condition weights are appropriately selected.Source: §2.2 · p. 8

2.3. Training algorithm

  • The optimization problem is non-convex, and the authors report that PINN errors depend strongly on the optimizer used.Their two-step procedure runs Adam for a prescribed number of iterations, then L-BFGS-B until the prescribed tolerance is met.Source: §2.3 · p. 9

3. Forward flow and advection-dispersion equations

This section evaluates PINN solutions for forward advection-dispersion problems across one- and two-dimensional settings, including coupled flow and heterogeneous conductivity. It examines agreement with analytical and numerical references and how accuracy varies with network size, loss weighting, and extrapolation beyond the training interval.

3.1. One-dimensional time-dependent ADE

  • For the 1D ADE at Pe = 62.8, the 4 × 40 PINN has relative error ϵ = 8.92 × 10−4, and errors stay below 1% across the considered cases.At Pe = 628, the 4 × 40 DNN fits the analytical solution, but a smaller 3 × 40 DNN has maximum point error close to 100% because of steep gradients near x = 1.Source: Figure 1 · p. 10, §3.1 · p. 12
  • At Pe = 628, a 4 × 40 DNN provides a perfect fit to the analytical solution, while the smaller 3 × 40 DNN has maximum point error close to 100%, as the solution develops very large gradients near x = 1.Source: Figure 2 · p. 11, §3.1 · p. 12
  • In 1D training, switching from Adam to L-BFGS-B reduces late-stage oscillations and increases convergence rate, while L-BFGS-B alone can become trapped in a high-loss local minimum.The final loss and iterations needed for a given tolerance increase with Pe; the two examples require nearly 5,000 more iterations at higher Pe.Source: §3.1 · p. 13

3.2. Two-dimensional time-dependent ADE: instantaneous Gaussian source

  • For the 2D ADE at Pe = 50 and 200, maximum point errors are below 3 × 10−3 for t ≤ 0.5 and below 0.01 at t = 0.6.Training enforces residuals and boundary conditions only through t = 0.5, so the t = 0.6 solution is an extrapolation.Source: Figure 4 · p. 14, §3.2 · p. 15, Figure 5 · p. 15
  • For Pe=50 and 200, w=10 gives the lowest relative L2 errors (1.49 × 10−3 and 2.64 × 10−3); at Pe=200, w=100 yields 4.11 × 10−3.The reported errors for w=1 are 2.41 × 10−3 at Pe=50 and 5.64 × 10−3 at Pe=200. Increasing boundary and initial-condition weights reduces boundary errors, but excessively large weights can diminish the PDE residual's influence.Source: §3.2 · pp. 15–16
  • The PINN outperforms the physics-informed extreme learning machine [41], greatly reducing its boundary errors.The authors attribute this improvement to larger weights on boundary- and initial-condition penalties.Source: §3.2 · p. 16

3.3. Steady-state ADE with a point source. Comparison with the finite element solutions

  • At Pe=100, PINN solutions closely agree with SUPG, while GFEM develops instabilities; PINN and SUPG are stable at both tested Pe values.Figure 7 compares the methods at Pe=1 and 100. SUPG stability depends on calibrating its stability parameter to mesh size and flow direction relative to the mesh.Source: Figure 7 · p. 17, §3.3 · pp. 17–18
  • PINN solutions show no oscillations at the dispersion front, and their accuracy is unaffected by collocation-point placement relative to flow direction.The authors attribute the absence of over- and undershoots to computing spatial derivatives analytically rather than by numerical discretization.Source: §3.3 · p. 18

3.4. Two-dimensional time-dependent ADE with a non-uniform velocity field

  • In the non-uniform-velocity example, coupled ADE and Darcy equations use a velocity field given by the Darcy solution and an anisotropic dispersion tensor; PINN approximates K(x) with a DNN trained on a subset of K values.The parameters γ in the velocity DNN are frozen when training the ADE solution.Source: §3.4 · pp. 18, 19, 21
  • The PINN hydraulic-head solution agrees well with the reference, with maximum point errors below 0.09 and relative L2 error ϵh = 1.63×10−3.Source: §3.4 · p. 21, Figure 9 · p. 22
  • Concentration errors are largest at the plume front, where the gradient is greatest, and decrease over time as the plume diffuses.Source: §3.4 · pp. 22–23
  • The PINN concentration shows close qualitative agreement with STOMP, with relative L2 error 3.82 × 10−2 and maximum point errors below 0.03.The reported concentration fields are at t = 2, 6, and 20 min; the supplied passage does not specify which time corresponds to the stated relative L2 error.Source: §3.4 · p. 22, Figure 11 · p. 23

4. Backward ADEs

The PINN is applied to backward ADEs with uniform and non-uniform velocity fields. Errors are generally small, though they rise toward the earlier prediction times and are larger near a concentration discontinuity.

4.1. Backward ADE with uniform velocity field

  • Backward diffusion problems are normally ill-posed, and standard numerical methods require regularization to solve them.The PINN uniform-velocity examples evaluate Pe = 1, 10, and 200, with maximum point errors of 8.07 × 10^-3, 1.23 × 10^-2, and 1.40 × 10^-2, respectively.Source: §4.1 · p. 23, Figure 12 · p. 24
  • For backward problem (29), relative L2 errors across the considered times (t = 0.0, 0.1, 0.3, 0.5) and Pe values (1, 10, 200) increase toward t = 0 but remain below 2 × 10^-2.The approximation errors slightly increase with Pe and appear related to gradients in the solutions.Source: §4.1 · p. 24

4.2. Backward ADE with non-uniform velocity field

  • The non-uniform-velocity backward problem assumes the concentration field at t = 10 min is known and seeks the field at earlier times.Its reference is the STOMP solution of the corresponding forward ADE.Source: §4.2 · p. 25
  • For non-uniform velocity, maximum point errors stay below 5 × 10^-2 for t > 0, but reach about 0.963 at t = 0 near the x = 0 boundary.The t = 0 error is attributed to a solution discontinuity at that boundary and the reference solution’s initial zero concentration.Source: §4.2 · pp. 25–26

5. Data assimilation

The PINN loss can incorporate concentration measurements alongside the backward ADE constraints. In the considered case, adding 40 measurements reduced prediction errors by more than 50% for 0 < t < 4 min.

5. Data assimilation

  • Concentration measurements can be incorporated by adding a measurement-loss term to the PINN objective.The term compares measured values with PINN predictions at the measurement locations and times.Source: §5 · pp. 26–27
  • For backward equations, the loss includes a mean-square terminal-condition mismatch in place of the forward ADE initial-condition term.The authors also state that state measurements can stabilize and regularize unstable or non-unique solutions.Source: §5 · pp. 26–27
  • Adding 40 measurements of u at 40 spatial locations randomly distributed in Ω and uniformly distributed in time reduced prediction errors by more than 50% for 0 < t < 4 min.Source: §5 · p. 27

6. Conclusion

The conclusion presents a PINN formulation for coupled Darcy flow and advection-dispersion with space-dependent conductivity, tested on one- and two-dimensional forward and backward ADEs. It reports agreement with numerical solutions, favorable comparisons for advection-dominated transport, and accurate backward solutions that improve when concentration measurements are incorporated.

6. Conclusion

  • The formulation solves coupled Darcy and ADE equations using a weighted residual loss, with initial- and boundary-condition residuals weighted more heavily than PDE residuals for accurate solutions.It is tested across one- and two-dimensional forward and backward ADEs over a range of Pe.Source: §6 · pp. 27–28
  • The PINN solutions for hydraulic head and concentration agree well with numerical solutions for coupled Darcy flow and advection-dispersion with space-dependent hydraulic conductivity and velocity fields.Source: §6 · p. 28
  • For advection-dominated transport, PINN outperforms GFEM and is comparable in accuracy with SUPG, which requires calibration of a stability parameter specific to the mesh and flow direction.The conclusion also reports that the PINN formulation outperforms earlier PINN studies [40] [41].Source: §6 · p. 28
  • In most cases, backward-ADE relative errors stay under 5% compared with the reference concentration field.The conclusion reports this accuracy for the backward ADE results.Source: §6 · p. 28
  • When available, concentration measurements can be incorporated into the PINN method and improve backward-ADE solution accuracy by more than 50% in the considered cases.The conclusion describes measurement incorporation as easy.Source: §6 · p. 28

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