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

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

physics.flu-dynMar 6, 2026

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

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

Organizations: Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, , United Kingdom

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.

Figures & tables

Boundary Condition	Equation
Inlet Velocity	$\mathbf{U}=\left(0.01,0\right)\quad\text{on the inlet boundary}$
Outlet Pressure	$P=0\quad\text{on the outlet boundary}$
Wall (No-Slip)	$\mathbf{U}=(0,0)\quad\text{on the wall boundaries}$

Table 1: Boundary conditions used in the simulation

Figure 1: Computational domain for the finite element analysis (FEM), showing the boundary conditions and design parameters used in the optimization process.

Parameter	Distribution
$p_{SIMP}$	$\mathcal{N}(0.01,0.001)$
$\alpha_{\text{max}}$	$\mathcal{N}(10000,1000)$
$q_{a}$	$\mathcal{N}(100,10)$
$w$	$\mathcal{N}(0.5,0.1)$

Table 2: Cold Plate Design Parameters

Figure 2: Model architectures illustrations for $\mathcal{M}_{\textrm{AE}}$ , $\mathcal{M}_{\textrm{UNET}}$ , and $\mathcal{M}_{\textrm{FNO}}$

Model Name	Symbol	Parameter Number
Convolutional Autoencoder	$\mathcal{M}_{\textrm{AE}}$	30389123
U-Net	$\mathcal{M}_{\textrm{UNET}}$	31042499
Fourier Neural Operator	$\mathcal{M}_{\textrm{FNO}}$	29516675

Table 3: Model Parameter Information

Channel Name	Metric	$\mathcal{M}_{\textrm{AE}}$		$\mathcal{M}_{\textrm{UNET}}$		$\mathcal{M}_{\textrm{FNO}}$
Channel Name	Metric	$\mathcal{L}_{data}$	$\mathcal{L}_{PI}$	$\mathcal{L}_{data}$	$\mathcal{L}_{PI}$	$\mathcal{L}_{data}$	$\mathcal{L}_{PI}$
$P$	$R^{2}$	0.9561719	0.9719298	0.8836706	0.9464990	0.9663823	0.9697812
	MSE	0.0412227	0.0264015	0.1094139	0.0503205	0.0316193	0.0284223
	Relative $L^{2}$	0.2092571	0.1674784	0.3409388	0.2312260	0.1832587	0.1737473
$U_{x}$	$R^{2}$	0.9895563	0.9929830	0.9952475	0.9972704	0.9982153	0.9976720
	MSE	0.0103068	0.0069250	0.0046901	0.0026938	0.0017613	0.0022975

Table 4: Model Error Comparison

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

Figure 6: Gradual Material Variations: topologies with minor deviations and linear transitions between material properties emerging in the flow structure

Figure 7: Developing Channels: configurations with partially formed channels, representing significant directional flow pathways

Figure 8: Clustered Variations: topologies where heterogeneities start forming distinct clusters, introducing localized complexity

Figure 9: Mature Channel Networks: most evolved topologies, characterized by well-defined and interconnected channel structures

Panel of Figure 10 — Figure 10: Model predictions of $\mathcal{M}_{\textrm{AE-PI}}$ , $\mathcal{M}_{\textrm{UNET-PI}}$ , and $\mathcal{M}_{\textrm{FNO-PI}}$ for a testing dataset with different resolutions.

Resolution	FEniCS Solver	$\mathcal{M}_{\textrm{AE-PI}}$		$\mathcal{M}_{\textrm{UNET-PI}}$		$\mathcal{M}_{\textrm{FNO-PI}}$
Resolution	FEniCS Solver	CPU	GPU	CPU	GPU	CPU	GPU
128	17.8406	0.0142	0.0010	0.0343	0.0025	0.1089	0.0020
256	82.0864	0.0366	0.0011	0.1400	0.0028	0.1338	0.0021
400	212.1420	0.1057	0.0010	0.3125	0.0024	0.2292	0.0021

Table 5: Comparison of average calculation speed for one design topology in seconds

Figure 11: Calculation speed comparison between models on GPU

Porous Media Flow

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May 18, 2026J/K move · Enter open · S save

Physics-informed convolutional neural networks for fluid flow through porous media

Rafał Topolnicki, Paweł Dłotko, Maciej Matyka

Dioscuri Center in Topological Data Analysis, Institute of Mathematics, Polish Academy of Sciences, ul. ´Sniadeckich 8, 00-656, Warsaw, Poland · Institute of Experimental Physics, Faculty of Physics and Astronomy, University of Wrocław, pl. M. Borna 9, 50-204, Wrocław, Poland · Institute of Theoretical Physics, Faculty of Physics and Astronomy, University of Wrocław, pl. M. Borna 9, Wrocław, 50-204, Poland+1

PDE Surrogate Modeling Physics-Informed ML Convolutional Neural Networks Computational Fluid Dynamics Porous Media Flow

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Accurate simulation of fluid flow in porous media is challenging due to complex pore-space geometries and the computational cost of solving the Navier-Stokes equations. This difficulty is particularly important when repeated simulations are required, as standard numerical solvers may converge slowly in intricate porous domains. We present a neural-network-based framework for predicting pore-scale velocity fields directly from sample geometry. The method uses a convolutional encoder-decoder architecture with skip connections to preserve spatial detail while extracting multi-scale features. Physical consistency is encouraged through a custom loss function combining velocity reconstruction with incompressibility, no-flow conditions inside solids, periodicity constraints, and agreement with the global tortuosity index. We analyze the influence of the corresponding loss weights and quantify the contribution of individual loss components to prediction accuracy. Several CNN backbones are evaluated to identify architectures providing accurate and robust predictions. The generalization ability of the trained model is tested on samples outside the training distribution, including changes in obstacle geometry, boundary conditions, porosity, and realistic porous structures. Finally, we demonstrate a practical use of the predicted velocity fields as initial conditions for Lattice-Boltzmann simulations. This warm-start strategy accelerates solver convergence, reducing the number of iterations in over 90% of tested cases.

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Porous Media Flow

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