To build faster and more energy-efficient computers, researchers need to understand nanoscale magnetic materials. But looking inside these tiny magnets is surprisingly difficult: modern microscopes only measure the magnetic field outside the material, not the magnetic pattern inside. Because many different internal patterns can produce the exact same field, working backwards has long been a challenge. Physicists at the University of Vienna, together with colleagues in Dresden, Singapore, and Ulsan, have now solved this problem by building the laws of physics directly into modern machine learning methods. Their method reveals the realistic magnetic structure and even determines the precise distance between the sensor and the sample. The study is published in Physical Review Letters, and the software is freely available as open source.
Magnetic materials only a few atomic layers thick could play a key role in future data storage and Energy efficient electronics. To develop them, scientists use advanced instruments like diamond Quantum sensors or magnetic force microscopes to scan their surfaces. Yet these tools cannot see the magnet directly. They only record the magnetic field extending into the space around the sample.
"Trying to reconstruct the magnet from this external field is like trying to guess the shape of a light bulb from the beam of light on a wall without knowing how far away it is held," says Alexander Setescak, doctoral researcher at the University of Vienna and first author of the study. "Countless different internal arrangements can produce virtually identical fields outside."
The Vienna-led team solved this challenge by combining modern machine learning methods with the fundamental laws of magnetism. Rather than guessing, their physics-informed algorithm evaluates possible magnetic patterns and selects the state that minimizes the material’s physical energy while matching the measured field. Patterns that violate basic physical laws are automatically ruled out.
"We let the physical energy of the magnet decide which solution is realistic," says Claas Abert, who led the study at the University of Vienna. "The algorithm finds the state the material would naturally form in nature."
The researchers tested their approach on measurements from the Max Planck Institute for Chemical Physics of Solids in Dresden. The team examined samples of an ultrathin magnet, Fe 3 − x GaTe 2 , synthesized at the National University of Singapore. To record the magnetic field, the researchers used a diamond sensor based on an atomic defect known as a nitrogen-vacancy center.
In these experiments, the sensor sits just tens of nanometres above the magnetic layer. Because magnetic fields weaken rapidly with distance, even a slight error in estimating this height distorts the reconstructed magnetic pattern. The team’s method solves this problem directly: because the field decay obeys strict physical laws, the software calculates both the magnetic pattern and the sensor height simultaneously, finding a distance of 81 nanometres.
"In experiments, the exact sensor height is usually unknown," says Uri Vool, group leader at the Max Planck Institute for Chemical Physics of Solids in Dresden. "Extracting the distance directly from the magnetic measurement removes a major source of error and makes the reconstructed map far more reliable."
Although demonstrated on diamond sensors, the approach is not limited to them. Because the method relies on fundamental laws of magnetic energy, the same principle can be adapted to other magnetic imaging techniques. To calculate this physical energy, the software uses NeuralMag, an existing open source micromagnetic package developed in Vienna. The reconstruction code itself is also freely available.
"Our hope is that other laboratories will pick up the framework and apply it to their own measurements," says Setescak. "Sharing these tools openly can help the community explore new magnetic materials for future information technology.
The Vienna team is already preparing the next step, extending the approach to 3D X-ray imaging of complex magnetic nanostructures.
Key messages
Physical Review Letters
Fourier-Space Approach to Physics-Informed Magnetization Reconstruction from Nitrogen-Vacancy Measurements
14-Oct-2026