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Prediction of Steady-State Flow through Porous Media Using Machine Learning Models

Jinhong Wang, Matei C. Ignuta-Ciuncanu, Ricardo F. Martinez-Botas, Teng Cao

TL;DR

This study develops a machine-learning framework for predicting steady-state flow through porous media governed by the Navier-Stokes-Brinkman equations, and demonstrates that FNO outperforms AE and U-Net, achieving a mean squared error as low as 0.0017 while providing speedups of up to 1000 times compared to CFD.

Abstract

Solving flow through porous media is a crucial step in the topology optimisation of cold plates, a key component in modern thermal management. Traditional computational fluid dynamics (CFD) methods, while accurate, are often prohibitively expensive for large and complex geometries. In contrast, data-driven surrogate models provide a computationally efficient alternative, enabling rapid and reliable predictions. In this study, we develop a machine-learning framework for predicting steady-state flow through porous media governed by the Navier-Stokes-Brinkman equations. We implement and compare three model architectures-convolutional autoencoder (AE), U-Net, and Fourier Neural Operator (FNO)-evaluating their predictive performance. To enhance physics consistency, we incorporate physics-informed loss functions. Our results demonstrate that FNO outperforms AE and U-Net, achieving a mean squared error (MSE) as low as 0.0017 while providing speedups of up to 1000 times compared to CFD. Additionally, the mesh-invariant property of FNO emphasizes its suitability for topology optimisation tasks, where varying mesh resolutions are required. This study highlights the potential of machine learning to accelerate fluid flow predictions in porous media, offering a scalable alternative to traditional numerical methods.

Prediction of Steady-State Flow through Porous Media Using Machine Learning Models

TL;DR

This study develops a machine-learning framework for predicting steady-state flow through porous media governed by the Navier-Stokes-Brinkman equations, and demonstrates that FNO outperforms AE and U-Net, achieving a mean squared error as low as 0.0017 while providing speedups of up to 1000 times compared to CFD.

Abstract

Solving flow through porous media is a crucial step in the topology optimisation of cold plates, a key component in modern thermal management. Traditional computational fluid dynamics (CFD) methods, while accurate, are often prohibitively expensive for large and complex geometries. In contrast, data-driven surrogate models provide a computationally efficient alternative, enabling rapid and reliable predictions. In this study, we develop a machine-learning framework for predicting steady-state flow through porous media governed by the Navier-Stokes-Brinkman equations. We implement and compare three model architectures-convolutional autoencoder (AE), U-Net, and Fourier Neural Operator (FNO)-evaluating their predictive performance. To enhance physics consistency, we incorporate physics-informed loss functions. Our results demonstrate that FNO outperforms AE and U-Net, achieving a mean squared error (MSE) as low as 0.0017 while providing speedups of up to 1000 times compared to CFD. Additionally, the mesh-invariant property of FNO emphasizes its suitability for topology optimisation tasks, where varying mesh resolutions are required. This study highlights the potential of machine learning to accelerate fluid flow predictions in porous media, offering a scalable alternative to traditional numerical methods.
Paper Structure (24 sections, 17 equations, 11 figures, 5 tables)

This paper contains 24 sections, 17 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Physical Problem Setup
  4. Case Setup
  5. Governing Equations and Physical Constraints
  6. Machine Learning Model Architecture
  7. Convolutional Autoencoder $\mathcal{M}_{\textrm{AE}}$
  8. U-NET $\mathcal{M}_{\textrm{UNET}}$
  9. Fourier Neural Operator $\mathcal{M}_{\textrm{FNO}}$
  10. Physics-Informed Learning and Training Strategy
  11. Physics-Informed Loss Construction
  12. Dataset Generation and Preprocessing
  13. Training Configuration
  14. Results
  15. Model Performance
  16. ...and 9 more sections

Figures (11)

  • Figure 1: Computational domain for the finite element analysis (FEM), showing the boundary conditions and design parameters used in the optimization process.
  • Figure 2: Model architectures illustrations for $\mathcal{M}_{\textrm{AE}}$, $\mathcal{M}_{\textrm{UNET}}$, and $\mathcal{M}_{\textrm{FNO}}$
  • Figure 3: Physics residuals $\mathcal{R}_{continuity}$ and $\mathcal{R}_{momentum}$ for models that are trained with and without the physics-informed loss term $\mathcal{L}_{physics}$
  • Figure 4: Comparison of predictions for different material field inputs across the $P$, $U_x$, and $U_y$ channels using three physics-informed machine learning architectures: autoencoder (AE), U-Net, and Fourier Neural Operator (FNO).
  • Figure 5: Uniform Topologies: most homogeneous configurations, with minimal variation in material distribution
  • ...and 6 more figures