Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Self-adaptive physics-informed neural network for forward and inverse problems in heterogeneous porous flow

Md. Abdul Aziz, Thilo Strauss, Muhammad Mohebujjaman, Taufiquar Khan

TL;DR

This work introduces a self-adaptive physics-informed neural network (PINN) framework for forward Darcy flow and inverse permeability inversion in heterogeneous porous media. It leverages region-aware, binary masks to represent piecewise-constant permeability and a trainable loss weighting scheme, paired with an interleaved AdamW-L-BFGS optimizer to achieve stable, robust training. Numerical results validate accurate forward surrogates and reliable recovery of the two-region permeability from indirect flow data, matching finite-element references. The approach offers a mesh-free, data-informed solver for Darcy flow with sharp permeability jumps and supports robust inversion under limited data conditions.

Abstract

We develop a self-adaptive physics-informed neural network (PINN) framework that reliably solves forward Darcy flow and performs accurate permeability inversion in heterogeneous porous media. In the forward setting, the PINN predicts velocity and pressure for discontinuous, piecewise-constant permeability; in the inverse setting, it identifies spatially varying permeability directly from indirect flow observations. Both models use a region-aware permeability parameterization with binary spatial masks, which preserves sharp permeability jumps and avoids the smoothing artifacts common in standard PINNs. To stabilize training, we introduce self-learned loss weights that automatically balance PDE residuals, boundary constraints, and data mismatch, eliminating manual tuning and improving robustness, particularly for inverse problems. An interleaved AdamW-L-BFGS optimization strategy further accelerates and stabilizes convergence. Numerical results demonstrate accurate forward surrogates and reliable inverse permeability recovery, establishing the method as an effective mesh-free solver and data-driven inversion tool for porous-media systems governed by partial differential equations.

Self-adaptive physics-informed neural network for forward and inverse problems in heterogeneous porous flow

TL;DR

This work introduces a self-adaptive physics-informed neural network (PINN) framework for forward Darcy flow and inverse permeability inversion in heterogeneous porous media. It leverages region-aware, binary masks to represent piecewise-constant permeability and a trainable loss weighting scheme, paired with an interleaved AdamW-L-BFGS optimizer to achieve stable, robust training. Numerical results validate accurate forward surrogates and reliable recovery of the two-region permeability from indirect flow data, matching finite-element references. The approach offers a mesh-free, data-informed solver for Darcy flow with sharp permeability jumps and supports robust inversion under limited data conditions.

Abstract

We develop a self-adaptive physics-informed neural network (PINN) framework that reliably solves forward Darcy flow and performs accurate permeability inversion in heterogeneous porous media. In the forward setting, the PINN predicts velocity and pressure for discontinuous, piecewise-constant permeability; in the inverse setting, it identifies spatially varying permeability directly from indirect flow observations. Both models use a region-aware permeability parameterization with binary spatial masks, which preserves sharp permeability jumps and avoids the smoothing artifacts common in standard PINNs. To stabilize training, we introduce self-learned loss weights that automatically balance PDE residuals, boundary constraints, and data mismatch, eliminating manual tuning and improving robustness, particularly for inverse problems. An interleaved AdamW-L-BFGS optimization strategy further accelerates and stabilizes convergence. Numerical results demonstrate accurate forward surrogates and reliable inverse permeability recovery, establishing the method as an effective mesh-free solver and data-driven inversion tool for porous-media systems governed by partial differential equations.

Paper Structure

This paper contains 17 sections, 1 theorem, 19 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Significance of the work
  3. Notations and mathematical preliminaries
  4. Variational formulations for the mixed FEM of the forward problem
  5. Theoretical justification of the proposed framework
  6. Identifiability of the parametric permeability model
  7. Self-adaptive loss weighting and gradient balancing
  8. Effect of Velocity Regularity on Inverse Permeability Identification
  9. Motivation for interleaved first- and second-order optimization
  10. Numerical validation and error estimation
  11. Finite element discretization and reference solutions
  12. Experimental configuration
  13. Results and discussions
  14. Forward PINN results
  15. Inverse PINN Results
  16. ...and 2 more sections

Key Result

Lemma 3.1

\newlabelidentifiability Let $\Omega=\Omega_1\cup\Omega_2$ with $\Omega_1\cap\Omega_2=\emptyset$ (up to a set of measure zero), and let $K(\boldsymbol{x})=K_1\chi_{\Omega_1}(\boldsymbol{x})+K_2\chi_{\Omega_2}(\boldsymbol{x})$ with unknown constants $K_1,K_2>0$. Assume $(\boldsymbol{u},p)\in (H^1(\O If for each $j\in\{1,2\}$ there exists a measurable set $E_j\subset \Omega_j$ with $|E_j|>0$ such th

Figures (10)

  • Figure 4.1: Schematic of the computational domain and imposed boundary conditions.
  • Figure 4.2: Forward PINN schematic diagram.
  • Figure 4.3: Inverse PINN schematic diagram.
  • Figure 5.1: Speed ($\|\boldsymbol{u}\|$) comparison between the forward PINN (left) and the FEM reference solution (right).
  • Figure 5.2: Pressure field $p(x, y)$ comparison between the forward PINN (Left) and FEM reference solution (Right).
  • ...and 5 more figures

Theorems & Definitions (2)

  • Lemma 3.1: Identifiability of region-specific constants $K_1,K_2$
  • proof