Property-variational Autoencoder (pVAE) for inverse design

The pVAE framework is trained on two datasets: a synthetic dataset of artificial porous microstructures and CT-scan images of volume elements from real open-cell foams. The encoder-decoder architecture of the VAE captures key microstructural features, mapping them into a compact and interpretable latent space for efficient structure-property exploration. The study provides a detailed analysis and interpretation of the latent space, demonstrating its role in structure-property mapping, interpolation, and inverse design. This approach facilitates the generation of new metamaterials with desired properties

Design Space Construction

1. Microstructures are represented in voxelized form (binary images).

2. The effective properties are:

   • Porosity $n^F$
   • Intrinsic permeability tensor $\mathbf{K}^{S}$

Porosity

The porosity can be calculated directly from the volume fraction of the pore phase in the binary image:

$$ n^F = \frac{V_\text{pore}}{V_\text{total}} $$

Intrinsic Permeability

Estimating the intrinsic permeability tensor $\mathbf{K}^S$ is more challenging and computationally expensive.

• A mesh-based approach can be used from the 3D binary image.
• The Lattice Boltzmann Method (LBM) discretizes the pore space in velocity space and solves the Navier–Stokes equations in a stochastic sense with a collision operator.
• From the LBM, we obtain the microscopic velocity field $\mathbf{u}(\mathbf{x})$.

The homogenized velocity is computed by volume averaging:

$$ \langle \mathbf{u} \rangle = \frac{1}{V_\text{pore}} \int_{V_\text{pore}} \mathbf{u}(\mathbf{x}) , dV $$

Applying Darcy’s law, the intrinsic permeability tensor is obtained as:

$$ \langle \mathbf{u} \rangle = -\frac{\mathbf{K}^S}{\mu} \nabla p $$

where:

• $\mu$ is the dynamic viscosity,
• $\nabla p$ is the macroscopic pressure gradient,
• $\mathbf{K}^S$ is the intrinsic permeability tensor.

Thus, $\mathbf{K}^S$ can be determined from the relationship between the averaged velocity and the applied pressure gradient. More details in [1].

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Reference:
[1] Nguyen Thien Phu, Uwe Navrath, Yousef Heider, Julaluk Carmai, Bernd Markert.
Investigating the impact of deformation on foam permeability through CT scans and the Lattice–Boltzmann method.
PAMM, 2023. [https://doi.org/10.1002/pamm.202300154]

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A surrogated model: Multiscale CNN-based intrinsic permeability prediction

To overcome the challenging and computationally expensive of numerical model, we applied a surrogate model that can be used to obtain the Intrinsic permeability faster. In particular, a CNN model is used to predict these values from the binary image instead of TPM-LBM. For more details, please check [2].

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Reference:
[2] Yousef Heider, Fadi Aldakheel, Wolfgang Ehlers. A multiscale CNN-based intrinsic permeability prediction in deformable porous media Applied Sciences, 15(5):2589, 2025.

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Variational autoencoder (VAE)

The Variational Autoencoder (VAE), introduced in 2013, is one of the most influential generative models.
It combines deep learning with probabilistic inference, enabling the mapping between high-dimensional data and a structured latent space. For more details, see [3].

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Reference:
[3] Kingma & Welling (2013), Auto-Encoding Variational Bayes.

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1. KL Divergence

The Kullback–Leibler (KL) divergence measures how one probability distribution differs from another:

$$ D_\text{KL}(p \parallel q) = \sum_{x \in X} p(x) \log \frac{p(x)}{q(x)} = \sum_{x \in X} p(x) \log (p(x)) - \sum_{x \in X} p(x) \log (q(x)) $$

The first term is negative entropy (-H) and the second term is negative cross-entropy (-CE). $D_\text{KL} = CE -H$.

$$ D_\text{KL}(p \parallel q) = - \sum_{x \in X} p(x) \log \frac{q(x)}{p(x)} = - \mathbb{E}_{p} \left[ \log \frac{q(x)}{p(x)} \right] \text{ with } \mathbb{E}(f(x)) = \sum_{x \in X} p(x)f(x) $$

Jensen inequality:

• $\mathbb{E}[f(x)] \geq f(\mathbb{E}(x)) \rightarrow \text{convex function}$
• $\mathbb{E}[f(x)] \leq f(\mathbb{E}(x)) \rightarrow \text{concave function}$

$$ - \mathbb{E}_{p} \left[ \log \frac{q(x)}{p(x)} \right] \leq - \log \left[ \mathbb{E}_{p} \left[ \frac{q(x)}{p(x)} \right] \right] $$

The right side:

$$ - \log \left[ \mathbb{E}_{p} \left[ \frac{q(x)}{p(x)} \right] \right] = - \log \left[ \sum_{x} p(x) \left[ \frac{q(x)}{p(x)} \right] \right] = - \log (1) = 0 $$

From that, we have:

$$ D_\text{KL} = - \sum p(x) \log \frac{q(x)}{p(x)} \leq 0 \Rightarrow D_\text{KL} \geq 0 $$

The KL divergence must be equal or larger than zero.

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2. Variational Inference

Bayes theorem:

$$ p(z|x) = \frac{p(x|z)p(z)}{p(x)} $$

Direct computation of the posterior $p(z|x)$ is often intractable.
Instead, VAE introduces a variational distribution $q_\phi(z|x)$ parameterized by a neural network (the encoder).

The true log-likelihood is:

$$ \log p_\theta(x) = \log \int p_\theta(x,z)dz = \log \int p_\theta(x|z)p(z)dz $$

Since this integral is intractable, we approximate it using variational inference, where the encoder learns $q_\phi(z|x)$.

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3. Driving the ELBO Loss

A VAE learns stochastic mapping between an observed $\mathbf{x}$-space (true space), whose empirical distribution $q_{\mathcal{D}}(\mathbf{x})$ is typically complicated, and a $\mathbf{z}$-space (latent space), whose distribution can be relatively simple. The generative model learns a joint distribution $p_{\mathbf{\theta}}(\mathbf{x}, \mathbf{z})$ that is often factorized as $p_{\mathbf{\theta}}(\mathbf{x}, \mathbf{z}) = p_{\mathbf{\theta}}(\mathbf{z}) p_{\mathbf{\theta}}(\mathbf{x}|\mathbf{z})$, with a prior distribution over latent space $p_{\mathbf{\theta}}(\mathbf{z})$, a stochastic decoder $p_{\mathbf{\theta}}(\mathbf{x}|\mathbf{z})$. The stochastic encoder $q_{\mathbf{\phi}}(\mathbf{z}|\mathbf{x})$, also called inference model, approximates the true but intractable posterior $p_{\mathbf{\theta}}(\mathbf{z}|\mathbf{x})$ of the generative model.

For any choice of inference model, including the choice of variational parameters $\mathbf{\phi}$, we have:

• The log marginal likelihood can be expressed as an expectation

$$ \log p_\theta(\mathbf{x}) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \big[ \log p_\theta(\mathbf{x}) \big]. $$

• Since $\log p_{\theta}(\mathbf{x})$ is constant w.r.t. $\mathbf{z}$, we can rewrite

$$ \log p_\theta(\mathbf{x}) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[\log \frac{p_\theta(\mathbf{x}, \mathbf{z})}{p_\theta(\mathbf{z}|\mathbf{x})} \right]. $$

• Add and subtract $\log q_\phi(\mathbf{z}|\mathbf{x})$

$$ \log p_\theta(\mathbf{x}) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p_\theta(\mathbf{x}, \mathbf{z})}{q_\phi(\mathbf{z}|\mathbf{x})} \right] + \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{q_\phi(\mathbf{z}|\mathbf{x})}{p_\theta(\mathbf{z}|\mathbf{x})} \right]. $$

• The second term is the KL divergence

$$ D_{\text{KL}}\left(q_\phi(\mathbf{z}|\mathbf{x}) \parallel p_\theta(\mathbf{z}|\mathbf{x})\right) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{q_\phi(\mathbf{z}|\mathbf{x})}{p_\theta(\mathbf{z}|\mathbf{x})} \right]. $$

Thus:

$$ \log p_\theta(\mathbf{x}) = \underbrace{ \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p_\theta(\mathbf{x}, \mathbf{z})}{q_\phi(\mathbf{z}|\mathbf{x})} \right] }_{\mathcal{L}(\theta,\phi;\mathbf{x})\text{ (ELBO)}} + D_{\text{KL}}\left(q_\phi(\mathbf{z}|\mathbf{x}) \parallel p_\theta(\mathbf{z}|\mathbf{x})\right). $$

$$ \underbrace{ \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p_\theta(\mathbf{x}, \mathbf{z})}{q_\phi(\mathbf{z}|\mathbf{x})} \right] }_{\mathcal{L}(\theta,\phi;\mathbf{x})\text{ (ELBO)}} = \log p_\theta(\mathbf{x}) - D_{\text{KL}}\left(q_\phi(\mathbf{z}|\mathbf{x}) \parallel p_\theta(\mathbf{z}|\mathbf{x})\right). $$

We have that the KL divergence must be larger than zero, demonstrating before (Section 1 - KL divergence). Then,

$$ \underbrace{ \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p_\theta(\mathbf{x}, \mathbf{z})}{q_\phi(\mathbf{z}|\mathbf{x})} \right] }_{\mathcal{L}(\theta,\phi;\mathbf{x})\text{ (ELBO)}} \leq \log p_\theta(\mathbf{x}) $$

Expanding the joint $p_\theta(\mathbf{x}, \mathbf{z}) = p_\theta(\mathbf{x}|\mathbf{z})p_\theta(\mathbf{z})$, we get:

$$ \mathcal{L}(\theta,\phi;\mathbf{x}) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p_\theta(\mathbf{x}|\mathbf{z}) p_\theta(\mathbf{z})}{q_\phi(\mathbf{z}|\mathbf{x})} \right] $$

$$ \mathcal{L}(\theta,\phi;\mathbf{x}) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log p_\theta(\mathbf{x}|\mathbf{z})\right] + \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p_\theta(\mathbf{z})}{q_\phi(\mathbf{z}|\mathbf{x})} \right] $$

$$ \mathcal{L}(\theta,\phi;\mathbf{x}) = \underbrace{ \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log p_\theta(\mathbf{x}|\mathbf{z})\right]}_{\text{Reconstruction loss}} - \underbrace{ \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{q_\phi(\mathbf{z}|\mathbf{x})}{p_\theta(\mathbf{z})} \right]}_{D_\text{KL}\left(q_\phi(z|x) \parallel p(z)\right)} $$

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The Evidence Lower Bound (ELBO) is maximized to train the VAE:

$$ \log p_\theta(\mathbf{x}) \geq \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})}[\log p_\theta(\mathbf{x}|\mathbf{z})] - D_\text{KL}\left(q_\phi(\mathbf{z}|\mathbf{x}) \parallel p(\mathbf{z})\right) $$

The ELBO has two competing terms:

• Reconstruction term: encourages accurate data reconstruction via the decoder $p_\theta(\mathbf{x}|\mathbf{z})$
• Regularization term: forces $q_\phi(\mathbf{z}|\mathbf{x})$ to be close to the prior $p(\mathbf{z})$

Thus, the VAE balances reconstruction quality with latent space regularization.

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4. KL Divergence for Standard Normal

When the prior is chosen as $p(\mathbf{z}) = \mathcal{N}(\mu_{p}, {\sigma_{p}}^2)$ and the encoder outputs a Gaussian $q_\phi(\mathbf{z}|\mathbf{x}) = \mathcal{N}(\mu_{q}, {\sigma_{q}}^2)$, the KL divergence has a closed form:

$$ D_\text{KL}\left(\mathcal{N}(\mu_{q}, {\sigma_{q}}^2) \parallel \mathcal{N}(\mu_{p}, {\sigma_{p}}^2)\right) $$

$$ D_\text{KL}\left(q_\phi(\mathbf{z}|\mathbf{x}) \parallel p(\mathbf{z})\right) = \mathbb{E}_{q_\phi(\mathbf{z}|\mathbf{x})} \left[ \log \frac{q_\phi(\mathbf{z}|\mathbf{x})}{p(\mathbf{z})} \right] $$

Given a dataset with i.i.d data (discete variable):

$$ D_\text{KL}\left(q_\phi(\mathbf{z}|\mathbf{x}) \parallel p(\mathbf{z})\right) = \sum_{\mathbf{z}} \frac{1}{\sqrt{2\pi{\sigma_q}^2}} e^{-\frac{1}{2}{\left( \frac{\mathbf{x} - \mu_q}{\sigma_q} \right)}^2} \log \left[ \frac{ \frac{1}{\sqrt{2\pi{\sigma_q}^2}} e^{-\frac{1}{2}{\left( \frac{\mathbf{x} - \mu_q}{\sigma_q} \right)}^2}}{ \frac{1}{\sqrt{2\pi{\sigma_p}^2}} e^{-\frac{1}{2}{\left( \frac{x - \mu_p}{\sigma_p} \right)}^2}} \right] $$

$$ D_\text{KL}\left(q_\phi(\mathbf{z}|\mathbf{x}) \parallel p(\mathbf{z})\right) = \sum_{\mathbf{z}} \frac{1}{\sqrt{2\pi{\sigma_q}^2}} e^{-\frac{1}{2}{\left( \frac{\mathbf{x} - \mu_q}{\sigma_q} \right)}^2} \left( \log \frac{\sigma_p}{\sigma_q} - \frac{(\mathbf{x}-\mu_q)^2}{2{\sigma_q}^2} + \frac{(\mathbf{x}-{\mu_p})^2}{2{\sigma_p}^2} \right) $$

We have the Gaussian kernel here. So, the KL divergence can be expressed as:

$$ D_\text{KL} = \mathbb{E}_q \left( \log \frac{\sigma_p}{\sigma_q} - \frac{(\mathbf{x}-\mu_q)^2}{2{\sigma_q}^2} + \frac{(\mathbf{x}-{\mu_p})^2}{2{\sigma_p}^2} \right) $$

$$ D_\text{KL} = \log \frac{\sigma_p}{\sigma_q} - \frac{\mathbb{E}_q \left[ (\mathbf{x}-\mu_q)^2 \right]}{2{\sigma_q}^2} + \frac{\mathbb{E}_q \left[ (\mathbf{x}-{\mu_p})^2 \right]}{2{\sigma_p}^2} $$

$$ D_\text{KL} = \log \frac{\sigma_p}{\sigma_q} - \frac{1}{2} + \frac{1}{2{\sigma_p}^2} \mathbb{E}_q \left[ (\mathbf{x}-{\mu_p})^2 \right] $$

$$ D_\text{KL} = \log \frac{\sigma_p}{\sigma_q} - \frac{1}{2} + \frac{1}{2{\sigma_p}^2} \left[\mathbb{E}_q [{\underbrace{(\mathbf{x}-{\mu_q})^2}_{{\sigma_q}^2}}] + 2(\mu_q - \mu_p){\underbrace{\mathbb{E}_q \left[ \mathbf{x} - \mu_q \right]}_{0}} + (\mu_q - \mu_p)^2 \right] $$

$$ D_\text{KL} = \log \frac{\sigma_p}{\sigma_q} - \frac{1}{2} + \frac{{\sigma_q}^2 + (\mu_q - \mu_p)^2}{2{\sigma_p}^2} $$

With the prior as Standard normal, $p_{\mathbf{z}} \sim \mathcal{N} (\mu_p = 0, {\sigma_p}^2 = 1)$

$$ D_\text{KL} = \log \frac{1}{\sigma_q} - \frac{1}{2} + \frac{{\sigma_q}^2+ {\mu_q}^2}{2} $$

$$ D_\text{KL} = \frac{1}{2} \left[ \log \frac{1}{{\sigma_q}^2} -1 + {\sigma_q}^2+ {\mu_q}^2 \right] $$

For other kernel, we need to drive the KL divergence according to it.

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Copyright

© Prof. Dr.-Ing. habil. Fadi Aldakheel

Leibniz Universität Hannover

Faculty of Civil Engineering and Geodetic Science

Institut für Baumechanik und Numerische Mechanik (IBNM)

https://www.ibnm.uni-hannover.de

Coded by Phu Thien Nguyen with the help of Copilot 😃

Paper: Deep learning-aided inverse design of porous metamaterials

The authors are: Phu Thien Nguyen, Yousef Heider, Dennis Kochmann, Fadi Aldakheel

Project Structure

pVAE
├── src
│   ├── model.py           # Defines the VAE architecture
│   ├── train.py           # Contains the training loop for the VAE
│   ├── train_vae_tune.py  # Contains the ray tune framework for hyperparameter tuning
│   ├── evaluate.py        # Evaluates the performance of the trained VAE
│   ├── latent_extract.py  # Extract the latent space
│   ├── interpolation.py   # The sphearical interpolation (Slerp)
│   ├── inverse.py         # Inverse process with target properties
│   └── utils
│       └── data_loader.py # Utility functions for loading and preprocessing data
├── data
│   ├── data3D200          # Directory containing 3D BMP images ($200^3$) (email me: [email protected])
│   ├── data3D200.csv      # Effective Properties for 3D BMP images ($200^3$)
│   ├── syn-data.ipynb     # Notebook for creating synthetic data ($100^3$), random pore position is uniform distribution
├── requirements.txt       # Lists the required Python dependencies and data information
├── .gitignore             # Specifies files to be ignored by Git
└── README.md              # Documentation for the project

Setup Instructions

1. Clone the repository:

   git clone <repository-url>
   cd CNN-pVAE-
   

2. Install the required dependencies:

   pip install -r requirements.txt
   

3. Place your 3D BMP images in the data3d directory.

Usage

To train the VAE, run the following command:

python src/train.py

After training, you can evaluate the model using:

python src/evaluate.py

Model Overview

The pVAE consists of a variational autoencoder (VAE) and a regressor. The VAE includes an encoder that compresses input images into a latent space and a decoder that reconstructs images from this latent representation. In addition to minimizing the reconstruction loss and the Kullback-Leibler divergence, the latent space is used by a regressor to predict effective material properties directly from the encoded representations. This joint framework enables both image reconstruction and property prediction, facilitating structure-property mapping and inverse design. For more information and results, please check [4, 5].

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Reference:
[4] P.T. Nguyen, Y. Heider, D. Kochmann, and F. Aldakheel, Deep learning-aided inverse design of porous metamaterials, CMAME, (2025)

[5] P.T. Nguyen, B.-e. Ayouch, Y. Heider, and F. Aldakheel, Impact of Dataset Size and Hyperparameters Tuning in a VAE for Structure-Property Mapping in Porous Metamaterials, PAMM, (2025)

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License

This project is licensed under the MIT License. See the LICENSE file for more details.