🔬 condensed matter

Topological Data Analysis combined with Machine Learning for Predicting Permeability of Porous Media

This paper demonstrates that combining topological data analysis with machine learning effectively predicts the permeability of porous media by utilizing structural, topological, and network-based features extracted from synthetic and experimental data.

Original authors: Ebru Dagdelen, Catherin Neena Lalu, Aakash Karlekar, Manav Arora, Matthew Illingworth, Jonathan Jaquette, Linda Cummings, Lou Kondic

Published 2026-05-19

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Ebru Dagdelen, Catherin Neena Lalu, Aakash Karlekar, Manav Arora, Matthew Illingworth, Jonathan Jaquette, Linda Cummings, Lou Kondic

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, complex sponge made of thousands of tiny, randomly scattered balls. You want to know how easily water can flow through it. In the real world, this is like trying to figure out how fast oil moves through underground rock or how well a water filter works.

Traditionally, scientists have to run incredibly slow, heavy computer simulations to answer this. It's like trying to calculate the exact path of every single drop of water as it weaves through the sponge. It takes a long time and a lot of computing power.

This paper presents a clever shortcut: using Artificial Intelligence (AI) to guess the answer instead of calculating it.

Here is how they did it, broken down into simple steps:

1. Making the "Sponge" (The Data)

First, the researchers didn't use real rocks or sponges. They used a computer program to generate 1,000 fake sponges.

The Recipe: They randomly dropped spheres (like marbles) into a 3D box. The empty spaces between the marbles are where the water would flow.
The Ground Truth: For every single fake sponge they made, they ran the slow, heavy simulation to get the exact answer for how permeable (flow-friendly) it was. This gave them a "answer key" to check their AI later.

2. Describing the Sponge (The Features)

To teach the AI, they needed to describe each sponge in a way a computer could understand. They didn't just say "it's a sponge." They broke the description down into three different "languages":

The "Architect" Language (Structural Features): This is the basic geometry. How big are the marbles? How much empty space is there? How twisty are the paths? It's like measuring the size of the rooms and the width of the hallways in a house.
The "Mapmaker" Language (Network Features): Imagine turning the sponge into a subway map. The "stations" are the big empty pockets, and the "tracks" are the narrow tunnels connecting them. They counted how many stations there were, how many tracks connected them, and how far apart the stations were.
The "Shape-Observer" Language (Topological Features): This is the most unique part. They used a math tool called Topological Data Analysis (TDA).
- The Analogy: Imagine the sponge is a landscape. The AI doesn't just look at the height of the hills; it looks at the shape of the land. It counts how many separate islands there are, how many loops or rings exist in the paths, and how long those loops last before they close up. It's like counting how many donuts are in a pile of dough, regardless of how squished the dough is.

3. The AI Teacher (Machine Learning)

They fed all these descriptions (Architect, Mapmaker, and Shape-Observer) into a Neural Network (a type of AI brain).

The Training: They showed the AI the description of a sponge and its "answer key" (the exact flow rate).
The Learning: The AI looked for patterns. It learned, "Oh, when the loops in the Shape-Observer language are long and the tunnels in the Mapmaker language are short, the water flows fast."
The Test: Once trained, they gave the AI new sponges it had never seen before and asked it to guess the flow rate.

4. The Results: Speed vs. Accuracy

The paper compares three ways to get the answer:

The Old Way (Direct Simulation): Extremely accurate, but takes about 25 hours to process all 1,000 sponges.
The AI Way (Using all descriptions): Almost as accurate as the old way (only about 5% error), but it takes less than 5 seconds to process all 1,000 sponges.
The "Map Only" Way: If you only use the subway map (Network features), the AI is a bit less accurate, but still pretty good.
The "Shape Only" Way: If you only use the Shape-Observer (Topological) data, the AI is surprisingly accurate and very fast to compute.

The Big Takeaway

The paper claims that Topological Data Analysis (TDA) is a powerful tool. By looking at the "shape" and "connectivity" of the holes (like counting loops and islands), the AI can predict how fluid flows almost as well as the most detailed geometric measurements, but much faster.

In short: Instead of simulating every drop of water (which is slow), they taught a computer to look at the "shape of the holes" and the "map of the tunnels" to instantly guess how fast the water will flow. This could save scientists massive amounts of time when designing filters, studying oil reservoirs, or understanding groundwater.

Note: The paper focuses entirely on synthetic (fake) data and does not claim to have tested this on real-world clinical or industrial samples yet, though it suggests this is a promising direction for future work.

Technical Summary: Topological Data Analysis combined with Machine Learning for Predicting Permeability of Porous Media

Problem Statement
Predicting the permeability of porous media is a critical challenge in fields ranging from groundwater filtration to oil recovery. Traditional methods rely on laboratory experiments or solving partial differential equations (PDEs) via numerical simulations (e.g., using the Stokes equations). While accurate, these PDE-based simulations are computationally intensive, requiring fine-grained meshes and significant processing time, particularly for large or structurally complex samples. The authors posit that the increasing availability of digitized representations of porous media makes this problem well-suited for data-driven approaches, specifically Machine Learning (ML), to bypass the computational cost of direct simulation while retaining predictive accuracy.

Methodology
The study establishes a workflow to train an ML model to predict permeability using geometric, topological, and network-based descriptors derived from synthetic porous structures.

Data Generation:
- The authors generated 1,000 synthetic 3D porous structures using the Porous Microstructure Analysis (PuMA) software.
- These structures consist of randomly distributed solid spheres (diameters 5–15 voxels) within a $50 \times 50 \times 50$ voxel domain, targeting a porosity of 0.5.
- Ground Truth: Permeability for each structure was calculated directly using PuMA's built-in capabilities by solving the Stokes equations and averaging the diagonal components of the permeability tensor.
Feature Extraction:
The authors extracted three categories of features to serve as input variables for the ML model:
- Structural Features: Measurable characteristics of the void space geometry, including sphere diameters, void space diffusivity, specific surface area, and tortuosity. These were computed using PuMA's built-in functions.
- Topological Features: Derived using Topological Data Analysis (TDA), specifically Persistent Homology (PH). Two methods were employed:
  - Alpha Complex Approach: A point cloud was created by subsampling void pixels. $\alpha$ -complexes were constructed, and Persistence Diagrams (PDs) were generated to track the birth and death of connected components (dimension-0) and loops (dimension-1).
  - Euclidean Distance Transform (EDT) Approach: The signed Euclidean distance from each void pixel to the nearest solid boundary was computed. Superlevel filtration was applied to these values to generate PDs.
  - Scalar Conversion: The PDs were aggregated into scalar features, specifically Total Persistence (TP), which sums the lifespans (death minus birth) of all topological features.
- Network Features: The porous structures were simplified into pore networks using the SNOW2 algorithm (implemented in PoreSpy). This algorithm identifies void centers (nodes) and connecting throats (edges). Descriptors included network connectivity (closed triads, paths of length two, edge counts) and network distance metrics (edge lengths, neighbor distances, closeness centrality).
Machine Learning Model:
- A feedforward neural network was implemented using TensorFlow.
- Architecture: One input layer, three dense hidden layers (128 nodes each), and one output layer, totaling approximately 50,000 parameters.
- Training: The dataset was split into 700 training, 150 validation, and 150 test samples. The model was trained for 1,000 epochs using the Adam optimizer and Mean Squared Error (MSE) loss. The Rectified Linear Unit (ReLU) activation function was used to introduce nonlinearity.
- Objective: To predict the logarithmic permeability ( $\log_{10} k$ ) based on the extracted feature vectors.

Key Results
The study evaluated the model's performance using the Mean Absolute Percentage Error (MAPE) and computational time across different feature sets:

Accuracy:
- Using all descriptors (structural, topological, and network) yielded a MAPE of 4.70%.
- Structural features alone achieved a MAPE of 4.75%.
- Topological features (Total Persistence) achieved a MAPE of 4.72% (for $\alpha$ -complex) and 6.31% (for EDT).
- Network connectivity features alone resulted in a MAPE of 6.78%, while network distance features performed poorly with a MAPE of 11.60%.
- The results indicate that topological and structural features provide near-optimal predictive power, while network distance metrics contribute little useful information.
Computational Efficiency:
- Ground Truth Calculation: Computing permeability for 1,000 structures via PuMA took 92,480 seconds.
- ML Prediction: Once trained, the neural network predicted permeability for 1,000 structures in 4.7 seconds (using all features).
- Feature Extraction: Structural feature extraction took the longest (18,040 seconds), while network feature extraction was fastest (820 seconds). Topological feature extraction (EDT) was notably efficient at 312 seconds.

Significance and Claims
The paper claims that Topological Data Analysis (TDA) provides a useful set of features that, when combined with ML, yield meaningful results for permeability prediction. Specifically:

Validation of TDA: The study demonstrates that persistence homology measures are valid predictors of permeability, offering performance comparable to traditional structural descriptors.
Efficiency vs. Accuracy Trade-off: While structural features offer the best balance of error and extraction time, topological features (particularly via EDT) offer a "middle ground," providing accurate predictions with significantly lower computational overhead than full structural analysis or direct simulation.
Educational Utility: The authors emphasize that the work presents these techniques in an accessible manner using well-documented software, aiming to serve as a resource for educators and researchers introducing students to ML and TDA in materials science.
Modest Scope: The authors note that the current study is limited to synthetic data ( $50 \times 50 \times 50$ domains) and that the network simplification (SNOW2) involves some loss of accuracy. They suggest that while the current results are promising, future work should explore larger domains and experimental datasets to confirm statistical homogeneity and generalizability.

The paper concludes that ML models trained on these descriptors offer a highly efficient alternative to traditional simulation-based methods, drastically reducing computational costs while delivering reliable permeability estimates.