🔬 materials science

DeepCormack: Fermi surface tomography using model-based data-driven algorithms

The paper introduces DeepCormack, a hybrid model-based and deep-learning algorithm that significantly accelerates and improves the quality of 3D Fermi surface reconstructions from ACAR data by generating realistic synthetic training sets via singular value and dynamic mode decomposition, thereby reducing acquisition times from months to weeks.

Original authors: Georg F. B. Lovric, Bryn Drury, Carola-Bibiane Schönlieb, Stephen B. Dugdale, Ander Biguri

Published 2026-07-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Georg F. B. Lovric, Bryn Drury, Carola-Bibiane Schönlieb, Stephen B. Dugdale, Ander Biguri

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 are trying to figure out the shape of a hidden, invisible city inside a block of metal. This isn't a city of buildings, but a "city" of electrons, the tiny particles that carry electricity and heat. Scientists call the edge of this electron city the "Fermi surface." Knowing its exact shape is like having a map of a treasure chest; it tells us why a metal conducts electricity, why it might become a superconductor, or why it acts like a magnet. To see this invisible map, scientists use a clever trick involving anti-matter. They shoot tiny particles called positrons into the metal. When a positron meets an electron, they vanish in a flash of light, shooting out two photons. By catching these photons from different angles, scientists can try to reconstruct the 3D shape of the electron city. However, this process is incredibly slow and blurry, like trying to draw a detailed portrait of a person while they are running away in the fog, and it often takes months to get a clear enough picture to be useful.

This is where a new team of researchers steps in with a tool they call "DeepCormack." Think of the old way of drawing this map as a very strict, rule-following artist who is great at math but terrible at guessing what the fog is hiding. They take the blurry photos and try to reconstruct the image using a method called the Modified Cormack Method (MCM). It works, but it's slow and the result is often fuzzy. The new paper introduces DeepCormack, which is like giving that strict artist a team of super-smart AI assistants. These assistants have studied thousands of "practice" maps and learned how to guess the missing details, remove the fog, and sharpen the lines. The researchers built a special system to create these practice maps using a computer simulation based on the element copper, because they couldn't wait months to get real photos for every test. They found that their AI-augmented method could produce a much clearer map in a fraction of the time, or even create a high-quality map from a much noisier, faster snapshot. However, they also discovered a catch: if the AI is trained only on copper, it sometimes gets confused when looking at a completely different metal, like ZrZn2, suggesting that the AI needs to be retrained with a specific "practice map" for every new metal it studies.

The core of the paper is about solving a difficult puzzle: how to turn a few blurry, noisy snapshots of electron momentum into a sharp, 3D picture of a material's Fermi surface. The researchers propose a family of algorithms that mix the traditional math-based method (MCM) with deep learning models, specifically Convolutional Neural Networks (CNNs), Multi-Layer Perceptrons (MLPs), and UNets. These neural networks act like different specialists in a repair shop. The CNN tries to clean up the noisy raw photos before they are even processed. The MLP refines the mathematical "radial" shapes that describe the density of electrons. Finally, the UNet looks at the final reconstructed image and fixes any remaining blurry spots or weird artifacts, essentially learning to "hallucinate" the correct details that the limited data missed.

To train these AI models, the team faced a huge problem: they didn't have enough real experimental data because collecting it takes months per sample. Instead, they invented a clever data generation method. They started with a perfect computer simulation of copper's electron map. Using a technique called Singular Value Decomposition (SVD), they broke this map down into its essential parts and then used Dynamic Mode Decomposition (DMD) to learn how those parts change as you move through the 3D volume. This allowed them to generate endless, realistic-looking synthetic 3D maps of copper that varied slightly from slice to slice, just like real data would. They then simulated the "fog" and "noise" of a real experiment by adding mathematical blur and random statistical noise to these perfect maps. This created a massive library of "ground truth" (perfect) images and their corresponding "noisy" versions, which the AI could learn from.

When they tested DeepCormack, the results were promising but nuanced. On the synthetic copper data (which the AI had seen during training), the new method was a massive improvement. At a standard count of 200 million electron events, the best DeepCormack model achieved a quality score (PSNR) of about 40.69 dB, compared to the old MCM method's 32.38 dB. Even more impressively, when they reduced the data to just 10 million counts (simulating a much faster, noisier scan), the DeepCormack model stayed strong at 38.22 dB, while the old method plummeted to 26.12 dB. This suggests that with DeepCormack, scientists could potentially cut their scanning time from months to weeks without losing image quality.

However, the paper also notes that the idea of a "one-size-fits-all" solution has limitations, though they are described as "moderate" rather than absolute. When they tested the models on a different metal, ZrZn2, which was not part of the training data, the performance of the best combined model (DCCMU) did drop, scoring 32.82 dB compared to MCM's 33.96 dB. This indicates that the AI had over-fitted to the specific features of copper. However, the paper highlights that this is not a total failure of the approach: the standalone UNet model (DCU) still outperformed the old MCM method on this new metal, scoring 37.44 dB. This suggests that while the complex combined models struggled to generalize, the core AI components still held value, and the authors note that Out-of-Distribution testing cannot be "completely discarded." The authors suggest that to fix this, scientists would need to run a quick computer calculation (Density Functional Theory) for the specific metal they are studying to generate a custom training set for the AI.

The paper concludes that while DeepCormack offers a significant leap forward in both speed and quality for Fermi surface tomography, it is not yet a fully autonomous solution for any material. The method works best when paired with a specific reference calculation for the material in question. The authors emphasize that their results are based on simulations and synthetic data, and while the noise simulation was carefully calibrated to match real experiments, testing on actual experimental data from different metals is the next necessary step. They do not claim to have solved the problem of Fermi surface reconstruction for all materials, but rather have demonstrated a powerful new tool that, when used correctly with the right training data, can dramatically accelerate the discovery of how metals behave.

Technical Summary: DeepCormack for Fermi Surface Tomography

Problem Statement
The experimental reconstruction of the 3D two-photon momentum density (TPMD) via Angular Correlation of Electron-Positron Annihilation Radiation (ACAR) is a critical method for studying material Fermi surfaces. Unlike surface-sensitive techniques, ACAR is bulk-sensitive, operates at room temperature, and does not require ultra-high vacuum or strong magnetic fields. However, reconstructing the 3D TPMD from a limited number of projections (typically 3–6) is a challenging inverse problem. Standard reconstruction relies on the Modified Cormack Method (MCM), which exploits crystal symmetries. Despite its theoretical robustness, the MCM suffers from poor signal-to-noise ratios (SNR) in experimental settings. Acquiring sufficient data (typically $\sim 10^8$ annihilation events per projection) for high-quality Fermi surface studies can take months per sample. Furthermore, instrumental smearing and noise, which vary systematically across the momentum density (degrading quality away from the center), often necessitate additional deconvolution. The authors posit that machine learning advances in medical image reconstruction (specifically Positron Emission Tomography, which shares similar geometry) could translate to Fermi surface tomography to improve reconstruction quality and reduce acquisition times.

Methodology
The paper introduces DeepCormack, a hybrid framework that integrates supervised deep learning models into the Modified Cormack Method (MCM). The approach is designed to correct noise, deconvolve instrumental smearing, and compensate for information loss inherent in limited angular sampling (5 projections).

Algorithmic Framework:
DeepCormack operates in three potential stages, each utilizing a different neural network architecture:
- 1D CNN (Projection Space): A residual convolutional neural network processes the raw 2D angular projections ( $f_{meas}$ ) to denoise and deconvolve them before reconstruction. It learns a correction $\Delta$ to the input projections.
- MLP (Radial Density Space): A Multi-Layer Perceptron refines the radial density functions ( $\rho$ ) derived from the MCM's Chebyshev coefficients. This stage operates in the space of the MCM's intermediate outputs.
- UNet (Image Space): A 2D UNet learns residual corrections on the final reconstructed 2D TPMD slices ( $\rho_{2\gamma}$ ). To handle the systematic variation in noise across slices (where counts decrease away from the center), the UNet can be conditioned on the total count level using Feature-wise Linear Modulation (FiLM).
The authors explore various configurations, including end-to-end training and pre-training individual components (CNN, MLP, UNet) before linking them.
Data Generation Strategy:
Recognizing the impracticality of collecting months of experimental data for training, the authors propose a data-driven simulation pipeline based on a single reference momentum density computed via Density Functional Theory (DFT) for Copper (Cu).
- SVD for Central Slices: Singular Value Decomposition (SVD) is applied to the Chebyshev coefficients of the reference Cu data to generate plausible, diverse central slices of the TPMD.
- DMD for 3D Evolution: Dynamic Mode Decomposition (DMD) is used to learn the underlying dynamics of how the momentum density evolves from slice to slice. A Koopman operator is computed to extrapolate the synthetic central slices into full, continuously varying 3D TPMD volumes.
- Simulation of Experimental Conditions: The synthetic 3D volumes are degraded to mimic real ACAR measurements. This includes applying detector resolution smearing (Gaussian convolution), Momentum Sampling Function (MSF) effects, and Poisson noise at various count levels (10M to 200M).
Validation:
The simulation pipeline was validated by comparing the degradation of simulated ZrZn2 data against real experimental ZrZn2 data. The authors found that adjusting the convolution widths and effective count levels in the simulation was necessary to match the degradation observed in real Fermi surface reconstructions.

Key Contributions

DeepCormack Framework: A novel family of algorithms integrating CNNs, MLPs, and UNets into the MCM pipeline to enhance Fermi surface reconstruction.
Synthetic Data Generation: A method using SVD and DMD to generate realistic, diverse 3D TPMD training data from a single DFT reference, overcoming the lack of large experimental datasets.
Operator Formulation: A derivation of the Modified Cormack Method in operator notation, facilitating its integration with modern inverse problem techniques.
FiLM Conditioning: The application of FiLM to condition the UNet on statistical precision (count levels), making the model robust to varying noise levels across the TPMD volume.

Results
The models were evaluated on three datasets: synthetic DMD data (in-distribution), Copper data (in-distribution but real DFT), and ZrZn2 data (out-of-distribution/OOD). Performance was measured using Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM).

Performance vs. MCM: DeepCormack models consistently outperformed the standard MCM. On synthetic data at standard counts (200M), the best configuration (pre-trained CNN + MLP $\to$ UNet) achieved ~40.69 dB PSNR compared to MCM's 32.38 dB.
Low-Count Robustness: DeepCormack demonstrated significant stability at reduced counts. At 10M counts, the best DeepCormack model (38.22 dB) maintained quality comparable to MCM at 200M counts (32.38 dB), whereas MCM performance dropped to 26.12 dB. This suggests acquisition times could be reduced by an order of magnitude without sacrificing reconstruction quality.
Generalization:
- Models performed well on data within the training distribution (DMD and Cu).
- Performance degraded significantly on the OOD ZrZn2 sample. The complex pre-trained configurations (e.g., CNN+MLP+UNet) performed worse than the baseline MCM on ZrZn2, indicating overfitting to the copper-derived training distribution.
- The standalone UNet (DCU) offered the best balance, outperforming MCM on ZrZn2 (37.44 dB vs 33.96 dB at 200M) and showing qualitative improvements in Fermi surface reconstruction when conditioned with FiLM.

Significance and Claims
The paper claims that DeepCormack offers a pathway to either significantly higher quality Fermi surface reconstructions or significantly faster acquisition times (reducing measurement duration from months to weeks). The authors emphasize that while the method currently relies on synthetic training data derived from a single reference (Copper), the workflow allows for the creation of sample-specific training data via DFT calculations. This "pre-computation" step (taking days) enables the subsequent experimental measurement to be performed with fewer counts while still yielding superior results to the standard MCM.

The authors remain modest regarding the generalization capabilities, acknowledging that the current models are sensitive to the training distribution and that performance on out-of-distribution samples (like ZrZn2) is limited. They suggest that future work should focus on training with broader DFT-generated datasets or real experimental data and developing loss functions based on Fermi surface metrics rather than image-space PSNR. The work aims to open Fermiology to the wider inverse problem community by demonstrating the viability of model-based, data-driven reconstruction in this domain.

Technical Summary: DeepCormack for Fermi Surface Tomography

More like this