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Predicting plasmonic pole representations of complex dielectric spectra using convolutional neural networks

This paper introduces a convolutional neural network trained on synthetic multipole-Padé spectra to efficiently and accurately invert complex dielectric spectra into their underlying plasmonic pole parameters, offering a robust, high-throughput alternative to conventional nonlinear fitting methods.

Original authors: Dario A. Leon

Published 2026-10-09

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CC BY 4.0

Original authors: Dario A. Leon

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

In the microscopic world of materials, electrons do not move in isolation; they swarm together, reacting to one another with a collective rhythm that defines how a substance interacts with light and electricity. When scientists study these materials, they often look at a "dielectric spectrum," a graph that acts like a fingerprint, revealing how the material screens out electric fields and how its electrons vibrate in response to energy. Within this complex curve lie hidden signatures of "plasmons," which are collective waves of electrons moving in unison. To understand the material's true nature, researchers need to extract specific details from these curves: the exact energy where the wave peaks, how quickly the wave fades away, and how much energy it carries. Traditionally, finding these hidden numbers has been a slow, difficult process, requiring scientists to guess and check their way through complex mathematical fitting until the curve matches their guess. This method is not only time-consuming but also fragile, often failing if the initial guess is slightly off or if the data is noisy.

A researcher at the University of Oslo has developed a new way to solve this puzzle, bypassing the slow guesswork entirely. Instead of trying to fit curves by hand or with standard algorithms, they trained a computer program, specifically a type of artificial intelligence known as a convolutional neural network, to recognize the patterns of these electron waves instantly. The researcher did not teach this computer by feeding it thousands of real-world examples from specific metals or semiconductors. Instead, they built a vast library of synthetic, or fake, spectra. They created these by taking the mathematical rules that describe how electron waves behave and randomly mixing the energy levels, fading rates, and strengths of the waves to generate millions of unique, theoretical curves. By training the computer on this endless variety of synthetic data, the network learned the fundamental relationship between the shape of the curve and the specific numbers that created it, rather than just memorizing the characteristics of a few specific materials.

When the researcher tested this trained network, it demonstrated a remarkable ability to generalize. The computer could look at a complex spectrum generated from real-world physics calculations or even from actual laboratory experiments and, in a single instant, identify the underlying electron waves with high precision. It successfully extracted the energy positions, the broadening, and the strength of the waves for materials ranging from simple metals like aluminum to more complex semiconductors like silicon and zinc oxide. The system worked even when the data contained overlapping waves that were difficult to separate or when the experimental measurements included small amounts of noise. In cases where the experimental data showed features that were not present in the synthetic training library, such as the sharp onset of a semiconductor or faint shoulder structures, the network still managed to capture the dominant features of the spectrum accurately, though it could not perfectly reconstruct every minor detail.

The study confirms that this approach offers a robust and efficient alternative to the traditional, labor-intensive methods of analysis. The researcher found that the network could learn the necessary mapping entirely from synthetic data, meaning it does not need to be retrained for every new material it encounters. This suggests that the same trained model can be applied to a wide variety of substances, from those studied in supercomputers to those measured in a lab, without needing material-specific adjustments. While the method is not yet perfect for every possible experimental quirk, the researcher shows that the limitations are systematic and can be addressed by simply expanding the library of synthetic examples to include more complex scenarios. This work establishes a new pathway for the rapid, automated analysis of how materials behave, turning a difficult, manual extraction process into a fast, reliable step that could accelerate the discovery and understanding of new electronic materials.

Technical Summary: Predicting Plasmonic Pole Representations of Complex Dielectric Spectra Using Convolutional Neural Networks

Problem Statement
Extracting spectral properties—specifically energy positions, broadenings, and spectral weights—from dielectric spectra is essential for interpreting collective electronic excitations in many-body physics. The inverse dielectric function, $\varepsilon^{-1}(\omega)$ , describes the dynamical screening of interacting electron systems and is central to methods like the GW approximation and the Bethe-Salpeter equation, as well as experimental probes such as electron-energy-loss spectroscopy (EELS).

The analytical structure of $\varepsilon^{-1}(\omega)$ in the complex-frequency plane can be represented compactly by a set of complex poles (plasmonic poles), where the real part defines the excitation energy, the imaginary part the broadening, and the residue the spectral weight. While multipole-Padé approximants (MPA) provide a powerful framework for constructing these representations, conventional approaches rely on nonlinear fitting procedures. These traditional methods are computationally demanding, highly sensitive to initialization, and difficult to automate for high-throughput analysis across diverse materials.

Methodology
The author develops a convolutional neural network (CNN) framework to perform the direct inversion of dielectric spectra into their underlying MPA pole structures. The core innovation lies in the training strategy: rather than training on a constrained database of real materials, the network is trained entirely on a synthetic ensemble generated from the analytical MPA expression with randomized parameters.

Synthetic Ensemble Construction: The training data consists of 20,000 spectra per pole count ( $n_p = 1$ to $5$), generated by randomly sampling pole positions ( $\text{Re } \Omega_p$ ), broadenings ( $\text{Im } \Omega_p$ ), and residues ( $r_p$ ) within physically motivated ranges. To ensure robustness, the ensemble includes "clean" spectra and "noisy" spectra containing additional weak MPA contributions (simulating experimental noise or unresolved features).
Inverse Problem Formulation: The network is trained to map the frequency-dependent response $Y(\omega)$ (or its imaginary part $\text{Im } Y(\omega)$ ) directly to the set of pole parameters $\{ \Omega_p, r_p \}$ . Two input configurations are tested: full complex response and imaginary-only response.
Network Architecture: A one-dimensional CNN with a fully connected regression head is employed. The architecture utilizes four convolutional layers with decreasing kernel sizes ( $k=101, 21, 7, 3$ ) to extract spectral features, followed by max-pooling and adaptive average-pooling. The output layer is parameterized to enforce physical constraints (e.g., positive residues, specific pole locations).
Training Objective: The loss function combines the error in predicted parameters ( $L_{par}$ ) with the error in reconstructing the input response ( $L_{res}$ ). This dual objective ensures that the network not only predicts parameters close to the ground truth but also generates a response that accurately reproduces the input spectrum.

Key Contributions

Synthetic-Only Training: The paper demonstrates that a CNN can learn the general mapping between spectral responses and their underlying multipole structures without requiring material-specific training data. This avoids biases introduced by the specific correlations found in real material datasets.
Robustness to Complexity and Noise: The framework is validated against spectra with multiple overlapping excitations and controlled spectral perturbations. The inclusion of "noise" in the training ensemble improves the model's ability to handle weak perturbations in target spectra.
Imaginary-Only Inversion: The study shows that training on only the imaginary part of the response ( $\text{Im } Y(\omega)$ ) yields results close to those obtained with the full complex response. This is significant for experimental applications where only the imaginary component is directly measurable, allowing for the reconstruction of the full pole representation without explicit Kramers-Kronig transformations.
Generalization to Real Data: The trained models successfully generalize to first-principles (RPA) calculations and experimental spectra of various materials (metals and semiconductors) that were not part of the training set.

Results

Synthetic Validation: On synthetic validation sets, the CNN accurately recovers pole positions, broadenings, and residues for spectra with up to five poles. The mean absolute errors increase monotonically with the number of poles ( $n_p$ ), reflecting the increased difficulty of resolving overlapping excitations, but remain within acceptable bounds for automated analysis.
First-Principles Application: The model successfully reconstructs RPA dielectric spectra for bulk Aluminum (Al), Calcium (Ca), and Titanium (Ti). For Al, a single pole suffices for the optical limit, while Ca and Ti require 3 and 4 poles, respectively. The reconstructed responses closely match the reference RPA data.
Experimental Application: The framework was applied to experimental spectra of Silicon (Si), Zinc Oxide (ZnO), Tantalum (Ta), and Molybdenum (Mo).
- For Si, Ta, and Mo, the CNN accurately captured the dominant spectral features using 1, 4, and 5 poles, respectively.
- For ZnO, a direct-band-gap semiconductor, the model reproduced the main peaks but struggled with the spectral onset and low-energy tail, which are not represented in the synthetic pole-based ensemble.
- The results indicate that while the model excels at capturing dominant plasmonic structures, it may miss secondary features (shoulders, onsets) or experimental artifacts not present in the synthetic training distribution.

Significance and Claims
The author claims that this work establishes a feasible framework for the high-throughput, automated analysis of dielectric spectra. By decoupling the training process from specific material databases, the approach offers a robust alternative to conventional nonlinear fitting. The study emphasizes that the method is a "proof of feasibility" rather than a fully optimized solution for all experimental spectral features.

The significance lies in the ability to extract physically interpretable pole parameters from complex spectra in a single forward pass. The author notes that the limitations observed (e.g., difficulty with spectral onsets or specific noise types) are systematic and can be addressed by expanding the synthetic ensemble to include a wider variety of spectral structures and noise profiles. This modularity suggests a pathway toward increasingly complex and accurate automated spectral analysis across diverse materials and applications.

Technical Summary: Predicting Plasmonic Pole Representations of Complex Dielectric Spectra Using Convolutional Neural Networks

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