Summary: A 2026 Nature Communications study study introduces a hybrid physics and machine learning framework for electron diffraction. Combining differentiable physical simulations with neural networks, the method achieves high accuracy crystal structure refinements and addresses long standing limits in modeling real experimental conditions.
Electron microscopy remains one of the most precise tools for resolving atomic scale structures. Techniques such as three dimensional electron diffraction allow researchers to reconstruct crystal structures from nanoscale samples that cannot be studied using traditional X ray methods. However a major limitation persists. While electron scattering is well described by theory, real experimental conditions introduce effects that are difficult to capture analytically. The new study proposes a hybrid framework that integrates physics based simulation with machine learning. Instead of relying solely on theory or purely data driven models, the method combines both. This ensures that simulations remain physically grounded while adapting to complexities observed in experiments. A central concept in the work is differentiable physics. In conventional simulation pipelines, calculating how outputs depend on parameters requires manually derived expressions, which becomes increasingly difficult for complex systems. Differentiable approaches avoid this by allowing gradients to be computed automatically across the full pipeline using backpropagation. This enables joint optimization. Structural parameters such as atomic positions and thermal motion can be refined at the same time as neural network components that represent experimental effects. The entire system becomes trainable end to end, allowing direct minimization of the difference between simulated and observed diffraction data.
The framework is applied to three dimensional electron diffraction. In this technique a crystal is rotated under an electron beam and diffraction patterns are collected at different orientations. These patterns contain detailed information about the atomic arrangement but are also influenced by factors such as thickness variation, defects, and beam interactions. To address this, the researchers embed a neural network within a physically accurate simulation based on the Bloch wave formalism. This formalism models how electrons scatter within a crystal including multiple scattering effects that are essential for quantitative interpretation. The neural network complements this by learning experimental variables that are difficult to model explicitly. A key advance lies in modeling crystal thickness. In real experiments thickness varies across the sample and changes with orientation, which strongly affects diffraction intensities. Traditional methods approximate this using simplified geometric assumptions, which often require manual tuning and may not reflect actual sample morphology. The hybrid approach replaces these assumptions with a data driven model. A neural network predicts how thickness varies with rotation angle, effectively learning the sample geometry directly from diffraction data. This allows the model to capture irregular and complex shapes without predefined constraints.
Because the system is fully differentiable, gradients propagate through both the physical simulation and the neural network. This allows efficient optimization using first order methods, avoiding the computational cost associated with second order techniques. As a result, large numbers of parameters can be refined simultaneously. The method was tested on synthetic and experimental datasets. In synthetic cases, the model accurately recovered atomic positions, thermal parameters, and thickness variations even when starting from imperfect initial conditions. This indicates strong stability and convergence. For experimental datasets including quartz, CsPbBr3, and paracetamol, the hybrid method achieved performance comparable to or better than established refinement approaches. In several cases it reduced residual errors, indicating improved agreement between simulation and experiment. The framework also correctly determined crystal chirality. Electron diffraction is sensitive to subtle asymmetries between mirror structures. The ability to recover the correct handedness confirms that the model preserves physically meaningful information.
Beyond performance, the approach offers conceptual advantages. It combines interpretability from physics with flexibility from machine learning. The physical model ensures consistency with known laws, while the neural network captures deviations arising from real experimental conditions. The framework can be extended to include additional effects such as beam damage, inelastic scattering, and detector noise. These factors currently limit accuracy in electron microscopy but can be incorporated into the same differentiable pipeline. There are broader implications for scientific computing. The work demonstrates how automatic differentiation can transform simulation workflows by enabling scalable optimization in complex systems. This approach is increasingly relevant across multiple areas of physics and engineering. Future directions include improving models of electronic structure. Current refinements often assume simplified charge distributions, which neglect subtle bonding effects. Hybrid approaches could incorporate more realistic descriptions and improve the interpretation of electron scattering data.
The framework is also adaptable to other microscopy techniques such as scanning transmission electron microscopy and related methods. This suggests a broader shift toward integrating machine learning directly within physical simulation pipelines. This study represents a methodological change. Instead of forcing experiments into simplified theoretical models, it allows models to adapt to experimental reality. By combining physics with learning, the framework provides both accuracy and flexibility, which are essential for advancing quantitative microscopy.
