DEPARTMENT OF COMPUTATIONAL AND DATA SCIENCES
Ph.D. Thesis Colloquium
Speaker : Nischal Karthik Mapakshi
S.R. Number : 06-18-00-10-12-19-1-17456
Title : “Structure-Preserving Physics-Informed Neural Networks for Modelling Flow in Anisotropic Porous Media with Pressure-Dependent Viscosity.”
Research Supervisor: Prof. Soumyendu Raha
Date & Time : June 19, 2025 (Thursday) at 03:30 PM
Venue : The Thesis Colloquium will be held on HYBRID Mode
# 102 CDS Seminar Hall /MICROSOFT TEAMS.
Please click on the following link to join the Thesis Colloquium:
MS Teams link
ABSTRACT
This work presents a comprehensive Physics-Informed Neural Network (PINN) framework to model subsurface flow through anisotropic porous media with pressure-dependent viscosity. The study begins with a strong-form PINN implementation, where the governing PDEs are directly encoded using automatic differentiation in TensorFlow. This avoids numerical quadrature and allows exact enforcement of physics through the strong form. To preserve physical realism—particularly non-negative pressures and the discrete maximum principle (DMP)—a custom sigmoid-based output transformation was introduced. This hard constraint mechanism enabled the network to achieve 0.0\% DMP violations across benchmark problems.
A key highlight of this study is the use of realistic physical parameters (permeability, viscosity, pressure) that span several orders of magnitude—something not commonly done in existing PINN literature due to conditioning issues. This is addressed using a robust non-dimensionalization strategy, which improves training stability and enhances the model’s applicability to real-world engineering problems.
Benchmarking against manufactured solutions and scaling studies revealed that while increasing collocation points improves accuracy up to a point, performance plateaus beyond a threshold, indicating limited returns. The effect of anisotropy and nonlinearity was also explored: while pressure-dependent viscosity had a mild effect on constraint satisfaction, directional anisotropy significantly increased the risk of DMP violations if constraints were not strongly enforced.
To address complex geometries such as annular domains, the study transitions to a Galerkin PINNs formulation, where the PDE is reformulated in weak form using test functions. This variational approach enhances stability, particularly in irregular geometries. Two constraint strategies were explored:
(1) Hard constraints using sigmoid or ReLU output transformations, and
(2) Soft constraints using interior-penalized loss terms to bound the pressure field.
The hard constraint formulation continued to deliver strict DMP preservation even in highly anisotropic regimes, while the soft version exhibited small, localized violations. A systematic sweep over the anisotropy parameter $\varepsilon$ was performed for both formulations. Results demonstrated a clear correlation between increasing $\varepsilon$ and DMP violation percentage. However, hard-constrained models consistently suppressed violations to near-zero levels. All pressure results were reported in dimensional units (Pa), and pressure and velocity fields were saved separately for each $\varepsilon$, ensuring transparency and reproducibility.
Hyperparameter optimization was also conducted using Bayesian methods. Results showed that increasing the number of layers or learning rate improves convergence up to a point, beyond which performance deteriorates due to overfitting or instability. Learning curves and error trends confirmed the presence of optimal regimes that balance accuracy and training efficiency.
Following the Galerkin formulation, the study now advances toward a Variational Multiscale (VMS) PINNs formulation. While traditionally used to separate coarse and fine scales, in this work VMS serves to enrich the variational residual by adding stabilizing terms on both sides, offering an alternative means of improving solution stability. As with the Galerkin formulation, both hard and soft constraint strategies will be explored for VMS, with an $\varepsilon$-sweep used to evaluate DMP adherence.
This unified PINN framework—spanning strong-form, Galerkin, and VMS formulations—demonstrates the ability to model nonlinear, anisotropic flow with high physical fidelity. By enforcing maximum principles and incorporating realistic parameter scales, the approach is well-suited for subsurface flow modeling in geophysics, petroleum recovery, and groundwater applications.
ALL ARE WELCOME
