Physics-informed automated surface reconstructing via low-energy electron diffraction based on Bayesian optimization

This paper presents a physics-informed Bayesian optimization framework that embeds multiple scattering LEED forward models to autonomously and efficiently solve the complex inverse problem of quantitative LEED-I(V) analysis for determining surface atomic structures.

The Gist

The Big Picture: Solving a 3D Jigsaw Puzzle Blindfolded

Imagine you are trying to figure out the exact shape of a complex 3D sculpture (like a tiny crystal surface) just by looking at the shadows it casts when a flashlight shines on it. This is essentially what scientists do when they study materials using Low-Energy Electron Diffraction (LEED).

They shoot electrons at a surface, and the electrons bounce off, creating a pattern of light and dark spots (a diffraction pattern). The pattern tells them where the atoms are. However, turning that pattern back into a 3D map of atoms is incredibly hard. It's like trying to guess the shape of a hidden object just by listening to the echo of a sound bouncing off it.

The Problem:
Traditionally, figuring out this "atomic map" has been like trying to tune a massive, old-fashioned radio with 50 different knobs. You have to guess which knobs to turn, how much to turn them, and in what order. If you turn the wrong knob, the signal gets worse. This process usually requires a human expert to spend hours or days manually tweaking settings, guessing, and hoping for the best. It's slow, prone to human error, and hard to repeat exactly.

The Solution:
This paper introduces a new, "smart" way to solve this puzzle using Bayesian Optimization (BO). Think of this as replacing the human radio tuner with a super-intelligent, self-driving robot that never gets tired and never makes a bad guess.

How the "Smart Robot" Works

The authors built a system that combines physics (the rules of how electrons bounce) with artificial intelligence (a smart search strategy). Here is how it works, step-by-step:

1. The "Trust Region" (The Search Radius)

Imagine you are looking for a lost key in a giant, dark field.

• Old Way: You might pick a spot, look around a tiny circle, and if you don't find it, you guess a new spot far away and start over.
• The New Way (Trust Region): The robot starts with a large search area. As it finds clues that get it closer to the key, it shrinks its search circle to focus intensely on that specific spot. If it hits a dead end (a "local trap"), it suddenly expands its search radius again to look in a completely different part of the field.
• In the Paper: The computer automatically adjusts how "wide" it looks for the solution. If the solution is getting better, it zooms in. If it gets stuck, it zooms out to find a better path.

2. The "Physics-Informed" Brain

Usually, AI needs to be fed millions of photos to learn what a cat looks like. But here, the robot already knows the laws of physics.

• Instead of guessing randomly, the robot uses a built-in "physics simulator" (like a video game engine that knows exactly how electrons behave).
• Every time the robot suggests a new atomic arrangement, the simulator instantly checks: "If the atoms were here, would the electron shadow look like the real experiment?"
• This means the robot doesn't need millions of training examples; it just needs to follow the rules of physics to find the answer.

3. The "Auto-Pilot" for Experiments

The system doesn't just find the structure; it also fixes the experimental mistakes.

• Sometimes the electron beam isn't perfectly aligned, or the sample is slightly tilted.
• The robot treats these errors as just another "knob" to turn. It automatically adjusts the angle of the beam and the vibration of the atoms until the shadow matches perfectly.

The Two Test Cases

The researchers tested their "Smart Robot" on two very different puzzles:

1. The Simple Puzzle (Silver Surface): A flat, simple surface with few atoms.

   • Result: The robot solved it quickly, finding the exact atomic positions and how much the atoms were vibrating (due to heat) without any human help. It found a solution that was just as good as the best human experts, but in a fraction of the time.

2. The Hard Puzzle (Iron Oxide Surface): A complex surface with 53 different "knobs" to turn (atoms moving in different directions, heat vibrations, beam angles).

   • Result: This is like trying to solve a Rubik's cube while blindfolded. The robot got stuck in a "local trap" (a solution that looked good but wasn't the best). However, its "Trust Region" strategy kicked in, expanded the search, and successfully jumped out of the trap to find the true best solution. It even corrected the angle of the electron beam automatically.

Why This Matters

• No More Guessing: You don't need a PhD in physics to tune the knobs anymore. The system does it all automatically.
• Reproducible: Because the robot follows a strict logic, two different people running the same experiment will get the exact same result.
• Faster: It cuts down the time needed to analyze complex materials from days to hours.
• Future-Proof: This method isn't just for electrons. It can be used for any scientific technique where you have to work backward from data to find a physical structure (like X-rays or microscopes).

The Bottom Line

This paper is about teaching computers to be better scientists than humans at a specific, difficult task. By combining the unchangeable laws of physics with a smart, adaptive search algorithm, they created a system that can automatically "reconstruct" the atomic world from experimental data. It's the difference between manually tuning a radio for hours and having a smart assistant that instantly finds the perfect station.

Technical Summary

1. Problem Statement

The Challenge of LEED-I(V) Analysis:
Low-energy electron diffraction (LEED) is a standard technique for determining surface atomic structures. However, the quantitative analysis of LEED intensity-energy curves (LEED-I(V)) is a complex inverse problem.

• Non-linearity: The relationship between surface structure and diffraction intensity is governed by highly non-linear multiple scattering processes.
• Current Limitations: Existing workflows (e.g., ViperLEED.calc) rely on sequential, expert-driven optimization. They require manual tuning of parameter ranges, step sizes, and optimization strategies. Structural parameters (atomic positions) and experimental parameters (beam angles, thermal vibrations) are often optimized in decoupled, sequential steps.
• Scalability Issues: These manual, heuristic approaches struggle with high-dimensional, complex parameter spaces, leading to issues with reproducibility, robustness, and the risk of getting trapped in local minima.
• Data-Driven Limitations: Purely data-driven methods (e.g., autoencoders) require massive training datasets and often lack physical interpretability.

2. Methodology

The authors propose a Physics-Informed Bayesian Optimization (BO) framework to solve the LEED-I(V) inverse problem autonomously.

• Core Formulation: The problem is cast as minimizing the Pendry R-factor ($R_P$), a metric quantifying the agreement between experimental and theoretical LEED-I(V) spectra.
  • $R_P(\Theta) = \frac{\sum \int (Y_{th} - Y_{expt})^2 dE}{\sum \int (Y_{th}^2 + Y_{expt}^2) dE}$
  • The forward model uses TensErLEED and ViperLEED packages to perform full multiple scattering simulations without linearization or gradient approximations.
• Bayesian Optimization (BO) Strategy:
  • Surrogate Model: A Gaussian Process (GP) with a Matérn 2.5 kernel models the relationship between input parameters ($\Theta$) and the R-factor.
  • Acquisition Function: Uses qUCB (for low dimensions) or Thompson Sampling (for high dimensions) to select the next candidate parameters.
  • Adaptive Trust Regions (TR): A key innovation is the use of dynamic trust regions.
    • If the R-factor improves significantly, the TR expands to encourage broad exploration.
    • If improvement stalls, the TR contracts to focus on local refinement.
    • This allows the algorithm to navigate non-convex landscapes and escape local minima without manual intervention.
• Warm-Restart Mechanism: To avoid getting stuck in local optima, the system employs a warm-start strategy. When a TR shrinks to a minimum size, the optimization restarts from new random seeds while retaining all previously simulated data points in the training set.
• Unified Parameter Space: Unlike traditional methods, this framework treats structural parameters (atomic coordinates), vibrational parameters (VIBROCC), and experimental parameters (incidence angle) as a single, coupled optimization space.

3. Key Contributions

1. Fully Automated Workflow: The method eliminates the need for human expertise in defining parameter ranges or optimization hierarchies. It achieves high-quality reconstruction through a single, unified automated task.
2. Physics-Informed Integration: By embedding the full multiple scattering forward model directly into the BO loop, the method ensures physical validity without relying on large training datasets or statistical correlations alone.
3. Simultaneous Optimization: It successfully optimizes coupled structural and experimental parameters (e.g., atomic positions + beam incidence angle + thermal vibrations) simultaneously, a task previously requiring sequential, decoupled steps.
4. Adaptive Exploration: The adaptive Trust Region mechanism effectively handles high-dimensional, non-convex search spaces, balancing global exploration and local exploitation dynamically.

4. Results

The framework was validated on two benchmark surfaces with varying complexity:

Case A: Ag(100)-(1×1) Surface (Low Dimensionality)

• Setup: 13 parameters (4 atomic coordinates + VIBROCC).
• Performance: The algorithm autonomously reduced the R-factor from an initial 0.2737 to a final 0.0772.
• Convergence: The process naturally evolved from global exploration (large TR) to local refinement (small TR).
• Physical Validation:
  • DFT Consistency: Structures with lower R-factors generally corresponded to lower DFT total energies, confirming physical plausibility.
  • VIBROCC Importance: Fixing thermal vibrations to bulk values caused the R-factor to skyrocket (0.4029), proving that simultaneous optimization of geometric and vibrational parameters is essential.
  • Degeneracy: The algorithm identified a narrow basin of nearly energy-degenerate solutions, demonstrating its ability to find the physically relevant confidence region.

Case B: Fe₂O₃(1102)-(1×1) Surface (High Dimensionality)

• Setup: 53 parameters (27 atomic coordinates, 2 VIBROCC types, and beam incidence angle $\theta$).
• Performance: The algorithm escaped a local optimum (stagnation at $R \approx 0.37$) by adaptively expanding the Trust Region (increasing the effective sampling radius).
• Final Result: Achieved a final R-factor of 0.1977 and correctly optimized the beam incidence angle to 0.8740° (compensating for experimental misalignment).
• Comparison: The results were comparable to or slightly better than manual ViperLEED.calc runs, with significantly less human effort.

5. Significance and Future Outlook

• Paradigm Shift: The work transitions surface characterization from an expert-driven, heuristic process to an autonomous, physics-driven paradigm.
• Generalizability: The framework is not limited to LEED. It can be extended to other inverse problems in characterization techniques (e.g., SXRD, STM) where a physics-based forward model exists.
• Multimodal Integration: The approach paves the way for combining multiple characterization techniques (e.g., LEED + SXRD) within a single optimization loop.
• Future Directions: The authors suggest integrating Machine-Learned Interatomic Potentials (MLIPs) to pre-filter energetically favorable structures, enabling the application of this method to even larger, more complex systems (e.g., twisted moiré superlattices) where DFT is computationally prohibitive.

In conclusion, this paper establishes a robust, scalable, and reproducible framework for solving complex inverse problems in materials science, demonstrating that physics-informed Bayesian optimization can autonomously determine surface structures with high fidelity.