🔬 materials science

Development of a Physics-Informed Neural Framework, MEOWN, for Rapid Prediction of Muon Stopping Sites in Crystalline Materials, for understanding Quantum Magnet employing Muon Spectroscopy

The paper introduces MEOWN, a physics-informed machine learning framework that rapidly and accurately predicts muon stopping sites in crystalline materials by combining electrostatic modeling with symmetry-driven optimization, offering a computationally efficient alternative to traditional density functional theory methods for studying quantum magnetism.

Original authors: A. Pandey, K. Sharma, S. Ghosh, G. Roy, T. Basu

Published 2026-09-16

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CC BY 4.0

Original authors: A. Pandey, K. Sharma, S. Ghosh, G. Roy, T. Basu

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

To understand the magnetic secrets hidden inside new quantum materials, scientists often use a tiny, fleeting particle called a muon. Imagine firing a stream of these positively charged particles into a solid crystal. They slow down almost instantly and come to rest at a specific, empty spot between the atoms of the crystal lattice. Once settled, the muon acts like a microscopic compass needle, spinning in response to the local magnetic field generated by the surrounding atoms. By watching how this spin changes over time, researchers can map out magnetic order, superconductivity, and other complex behaviors that are invisible to other tools. However, to read this map correctly, scientists must know exactly where the muon stopped. If they guess the wrong spot, their interpretation of the magnetic data becomes ambiguous, potentially leading to a misunderstanding of the material's true nature.

For decades, the standard way to find these stopping spots has been to use powerful computer simulations based on the laws of quantum mechanics. These calculations are incredibly accurate but also painfully slow and expensive, often requiring days of computing time for a single material. This bottleneck has made it difficult to quickly screen new materials or analyze data in real time. A team of researchers from India has now developed a new approach that bypasses this slowness without sacrificing accuracy. They created a tool called MEOWN, which combines the known laws of physics with a type of artificial intelligence to predict where a muon will stop in a crystal in just a few minutes.

The researchers built MEOWN to understand the forces that guide a muon to its resting place. They knew that the muon is attracted to negative charges in the crystal, repelled by electron clouds, and influenced by the way the crystal's atoms shift slightly to accommodate it. Instead of trying to calculate every single interaction from scratch, which is what the slow traditional methods do, MEOWN uses a "physics-informed" strategy. This means the software is not just guessing based on patterns; it is taught the fundamental rules of how electricity and atoms interact. The system takes a digital model of a crystal, fills it with these physical rules, and then uses a lightweight neural network—a simple form of artificial intelligence—to scan the entire structure. The network acts like a high-speed detector, highlighting the most likely regions where the muon would settle based on the energy landscape created by the physical forces.

To test if this fast method worked, the team applied it to a variety of well-known materials, including a metallic magnet called MnSi and several ionic crystals like calcium fluoride and lithium fluoride. In every case, the software successfully identified the correct stopping spots and calculated the magnetic fields at those locations. For example, in the material cobalt fluoride, the program predicted a magnetic field strength of 0.277 Tesla, which is extremely close to the 0.265 Tesla found by the slow, traditional computer methods and aligns well with actual experimental measurements. The software achieved this level of precision in less than a few minutes on a standard desktop computer, a task that would typically take much longer with conventional techniques.

The results show that MEOWN can serve as a rapid, reliable first step for scientists studying quantum materials. It does not replace the deep, detailed calculations needed for final confirmation, but it allows researchers to quickly narrow down the possibilities and focus their efforts on the most promising candidates. By embedding the laws of physics directly into the learning process, the team created a tool that is both fast and scientifically rigorous. This approach opens the door to analyzing a much wider range of materials and could accelerate the discovery of new quantum phenomena, turning a process that once took days into one that takes minutes.

Technical Summary: MEOWN Framework for Muon Stopping Site Prediction

Problem Statement
Muon-spin rotation/relaxation ( $\mu$ SR) is a critical microscopic probe for investigating magnetic order, spin dynamics, and superconductivity in quantum materials. However, the quantitative interpretation of $\mu$ SR data relies fundamentally on identifying the equilibrium muon stopping site within a crystal lattice. Traditionally, this identification is performed using Density Functional Theory (DFT) based structural relaxation. While accurate, conventional DFT approaches are computationally demanding, time-consuming, and require significant expertise, particularly for large supercells, low-symmetry crystals, or strongly correlated systems. Existing software tools (e.g., MuESR, muFinder) often inherit these high computational costs, limiting their utility for rapid data analysis and high-throughput materials screening.

Methodology: The MEOWN Framework
The authors introduce MEOWN (Muon Engine for Optimized Weighted Networks), a physics-informed machine learning framework designed to predict energetically favorable muon stopping sites with substantially reduced computational cost. The framework integrates first-principles physics with a lightweight deep learning architecture through the following components:

Physics-Informed Input Representation:
Instead of relying solely on geometric descriptors, MEOWN constructs a 3D voxel grid (48 $\times$ 48 $\times$ 48) containing four physics-based input channels derived from a Polarizable Unperturbed Electrostatic Potential (P-UEP) model:
- Negative total interaction energy ( $-V_{Total}$ ).
- Coulomb potential from ionic cores ( $1/q_i r_i$ ).
- Polarization potential ( $V_{Pol}$ ).
- Minimum-distance field ( $d_{min}$ ).
  The P-UEP model explicitly incorporates Hartree/exchange-correlation potentials, Thomas-Fermi electronic screening, Tang-Toennies polarization damping, Born-Mayer overlap repulsion, and quantum zero-point energy (ZPE) corrections.
Neural Network Architecture:
A lightweight 3D Convolutional Neural Network (3D-CNN) with fewer than 5,000 trainable parameters processes the input tensor. The network consists of four successive convolutional layers (channel progression: 4 $\to$ 16 $\to$ 32 $\to$ 16 $\to$ 1) with batch normalization and ReLU activation. The CNN outputs a 3D probability map indicating regions likely to host stable muon sites.
Hybrid Seeding and Optimization:
Candidate sites are generated via a multi-stage strategy combining:
- CNN Probability Maxima: High-probability regions from the network.
- Physics-Guided Seeds: High-symmetry interstitial positions (body-center, face-center, etc.) and chemically informed seeds (e.g., near electronegative anions, specific bridges like F–F in fluorides).
  These candidates are filtered by crystallographic symmetry and minimum distance criteria, then refined using L-BFGS-B optimization on the P-UEP potential energy surface.
Symmetry Constraints:
The framework explicitly utilizes crystallographic symmetry (via the spglib library) to reduce the search space. Energy gradients and Hessian matrices are projected onto the symmetry-allowed subspace to ensure relaxed sites remain consistent with the host crystal's space group.
Training and Implementation:
The model is trained using a self-supervised approach where the target potential is generated directly by the P-UEP physics engine, eliminating the need for manually labeled datasets. The loss function minimizes the Mean Squared Error (MSE) between the CNN-predicted potential and the physics-engine potential. The software is implemented in Python using PyTorch for the neural network and NumPy/SciPy for the physics engine, supporting both CPU and GPU execution.

Key Contributions

Development of MEOWN: A general-purpose software framework that combines a P-UEP physical model with a 3D-CNN to predict muon stopping sites.
Physics-Informed Learning: Explicit embedding of electrostatic, screening, polarization, and quantum mechanical interactions into the learning workflow, ensuring physical interpretability and consistency without relying on black-box data fitting.
Computational Efficiency: The framework achieves a significant speedup over conventional DFT methods, completing the entire prediction workflow (from CIF input to ranked sites) in a few minutes on a standard desktop computer.
Automated Workflow: A fully automated pipeline requiring only a Crystallographic Information File (CIF) as input, with no need for user-defined initial positions.

Results and Validation
The framework was validated against a diverse set of benchmark materials, including the metallic helimagnet MnSi and ionic fluorides CoF $_2$ , CaF $_2$ , LiF, and NaF.

Accuracy: Predicted stopping sites, Wyckoff positions, and local dipolar magnetic fields showed close agreement with previously reported experimental data and DFT+ $\mu$ $μ$ results.
- For CoF $_2$ , MEOWN predicted a local dipolar field of 0.277 T, compared to a DFT+ $\mu$ value of 0.265 T (3.5% deviation) and an experimental value of 0.228 T.
- For MnSi, the predicted site at fractional coordinates (0.55, 0.55, 0.55) with a field of 0.123 T matched established experimental and theoretical findings.
Transferability: The model successfully generalized across chemically distinct systems (metallic vs. ionic) without material-specific fitting parameters.
Speed: The entire process for benchmark materials was completed in less than a few minutes, contrasting with the hours or days typically required for repeated DFT structural relaxations.

Significance and Claims
The paper positions MEOWN not as a replacement for fully self-consistent first-principles calculations, but as a computationally efficient intermediate framework. Its primary significance lies in:

Rapid Screening: Enabling the rapid identification of physically plausible muon sites prior to expensive DFT calculations, facilitating high-throughput materials screening.
Accessibility: Providing a user-friendly tool that requires minimal expertise in DFT analysis, making $\mu$ SR data interpretation more accessible.
Physical Rigor: Demonstrating that embedding established physical principles into machine learning can yield interpretable, robust, and accurate predictions while maintaining scientific rigor.

The authors conclude that MEOWN offers a practical balance between computational speed and physical accuracy, serving as a valuable tool for the $\mu$ SR community in studying magnetic, superconducting, and complex quantum materials.

Technical Summary: MEOWN Framework for Muon Stopping Site Prediction

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