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Why an outside field cannot uniquely reveal the magnet inside
Magnetic microscopes measure the field outside a sample, while researchers often want the magnetization—the direction and strength of magnetic moments—inside the material. That is an inverse problem: the calculation runs backward from a field map to the internal source that produced it.
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The backward step is fundamentally ambiguous. Countless different internal arrangements can produce virtually identical fields outside an ultrathin sample. The problem becomes even less certain when the distance between the sensor and the material is not known precisely. Magnetic fields weaken rapidly with distance, so a small height error changes the apparent size and strength of structures in a reconstructed map.
As Alexander Setescak, a doctoral researcher at the University of Vienna and the study’s first author, put it, reconstructing the magnet 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.”
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How the physics-guided reconstruction works
1. Start with the measured field
The input is a magnetic-field map recorded above the ultrathin material. In the reported experiment, the field was measured with a diamond sensor using a nitrogen-vacancy (NV) center. The measurement contains real information about the magnet, but not a unique picture of its internal state.
2. Generate candidate magnetization states
The machine-learning workflow proposes possible internal magnetization patterns. It does not choose a result because it merely resembles a training image or looks visually smooth. For every candidate, the algorithm calculates the magnetic field that candidate should produce at the sensor.
3. Enforce agreement with the experiment
Candidates whose predicted fields disagree with the measured map are penalized. This data-fitting step keeps the solution tied to the actual NV measurement rather than to an unconstrained image-generation process.
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4. Use magnetic energy to select a realistic state
Several candidates can still fit the external field. The method therefore evaluates the magnet’s physical energy and favors states that minimize it while satisfying the field data. In the words of study leader Claas Abert, “We let the physical energy of the magnet decide which solution is realistic.” The selected pattern is intended to be a state the material could naturally form, not simply one of many mathematically possible answers.
5. Infer the sensor height at the same time
Sensor height is treated as an unknown in the optimization rather than as a fixed number supplied in advance. The algorithm searches for the distance and magnetization pattern that jointly provide a good field match and a physically plausible energy. This removes a major source of uncertainty that normally has to be calibrated separately.
What the Fe3−xGaTe2 demonstration found
The team analyzed Fe3−xGaTe2 synthesized at the National University of Singapore. Magnetic measurements were made at the Max Planck Institute for Chemical Physics of Solids in Dresden with an NV-center diamond sensor. Applying the reconstruction to those data produced an estimated sensor distance of 81 nanometres.
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That figure is the result reported for this particular material, measurement and reconstruction; it is not a universal working distance for NV magnetometry or a guaranteed accuracy level for every sample. Its importance is methodological: the same field data were used to infer both the internal magnetic pattern and the probe height.
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| Part of the problem | What the experiment provides | What the physics-guided workflow adds |
|---|---|---|
| Observed quantity | Magnetic field outside the ultrathin sample | A forward calculation that predicts the field for each candidate magnetization |
| Internal pattern | Not directly observed and not uniquely determined | Optimization over candidate states constrained by the measured field and magnetic energy |
| Sensor-to-sample distance | Often uncertain in an experiment | Estimated jointly with the magnetization |
| Interpretability | A field map alone does not identify the unique source | The chosen solution has an explicit field fit and physical-energy rationale |
| Evidence for comparative performance | Not established in the reported work | No head-to-head numerical benchmark against another inversion or black-box AI method was provided |
This does not make the reconstruction mathematically unique in every circumstance. It narrows the acceptable solutions by adding a model of how magnetic systems behave. If the assumed material properties, energy terms or measurement model are wrong, the result can still be misleading; the method makes those assumptions explicit rather than eliminating them.
Why estimating height from the same data matters
The external field decays quickly as the probe moves away from a sample. An incorrect height can therefore make a reconstruction assign the wrong scale or intensity to magnetic features. Uri Vool, who leads the Dresden group, said that exact sensor height is usually unknown in experiments and that extracting the distance directly from the magnetic measurement removes a major source of error.
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Joint estimation is especially useful for thin or nanoscale specimens, where a distance of a few tens of nanometres is a substantial fraction of the feature size being studied. The reported 81-nanometre estimate demonstrates the workflow’s ability to solve for that geometric parameter in the same optimization, but it does not by itself establish the uncertainty of the estimate.
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The workflow uses NeuralMag, an existing open-source micromagnetic package developed in Vienna. The reconstruction implementation is also described by the team as freely available. That combination could let other laboratories adapt the approach to their own field maps instead of rebuilding the magnetic forward model and optimization from scratch.
The researchers are preparing an extension for three-dimensional X-ray imaging of complex magnetic nanostructures. X-ray data introduce different measurement physics and reconstruction requirements, so that planned work should be understood as an extension under development, not as a demonstrated capability of the current ultrathin-magnet result.
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What researchers should check before applying it
- Measurement model: Confirm that the NV sensor response, field component and sample geometry used in the calculation match the actual experiment.
- Material model: Supply realistic magnetic parameters and energy terms for the material being reconstructed.
- Distance identifiability: Test whether the available field map contains enough spatial information to distinguish sensor height from changes in magnetization.
- Physical validation: Compare the inferred state with independent observations or known behavior when possible; a low data-fit error alone does not prove that the internal pattern is correct.
- Reproducibility: Record software versions, input fields, geometry and optimization settings when sharing a reconstruction.
What this result means for magnetic imaging
The central advance is not that AI can literally see through a magnet. It is that a learned optimizer can combine incomplete external observations with the governing physics to choose a defensible internal explanation. By solving for probe height alongside magnetization, the approach addresses a practical calibration problem at the same time as the inverse problem.
The University of Vienna-led team hopes other laboratories will apply the framework to their own measurements. Whether it improves accuracy over conventional inversion in a particular experiment will depend on the material model, sensor data and validation available; the reported study does not supply a numerical head-to-head benchmark. Its concrete contribution is a physics-constrained reconstruction workflow, an 81-nanometre distance estimate in Fe3−xGaTe2, and software described as open source for further testing.
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