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A University of Vienna-led team has shown that machine learning can infer an ultrathin magnet’s internal magnetization from measurements taken only outside the material. The method combines field calculations with magnetic-energy minimization, and it estimates the gap between the sample and the sensor at the same time. In a demonstration on Fe3−xGaTe2, it inferred a sensor distance of 81 nanometres.
The problem: an outside field does not uniquely reveal the inside
Magnetic microscopes measure a field above or around a sample, not the direction and strength of every magnetic moment inside it. Inferring the internal magnetization from that external map is an inverse problem: several different internal arrangements can produce almost the same field outside the material.
Alexander Setescak, a doctoral researcher at the University of Vienna and first author, compared the task with “trying to reconstruct the magnet from this external field … 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.” He also noted that “countless different internal arrangements can produce virtually identical fields outside.”
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The ambiguity is especially serious for ultrathin magnets. Their field falls rapidly with distance, so an error in the assumed sensor height changes the apparent size, strength and fine structure of features in the reconstructed map.
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How the physics-guided reconstruction works
1. Start with the measured field
The input is a spatial map of the magnetic field measured above the sample. In the reported experiment, the field came from a diamond sensor containing a nitrogen-vacancy (NV) center.
2. Generate candidate magnetization states
The algorithm proposes internal magnetization patterns and calculates the external field each pattern would produce. It then compares those calculated fields with the experimental map. A close field match is necessary, but it is not sufficient, because many candidates can fit the same data.
3. Enforce magnetic-energy constraints
Instead of selecting a solution solely because it resembles the measured image, the optimization penalizes states that are physically implausible. It uses the magnet’s energy to favor arrangements that a real material could naturally form while still matching the observed field.
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Study leader Claas Abert described the principle this way: “We let the physical energy of the magnet decide which solution is realistic.” The result is a physics-guided machine-learning workflow rather than a black-box image-to-image prediction.
4. Fit the sensor height jointly
The reconstruction treats the sensor-to-sample distance as an unknown parameter, optimizing it together with the magnetization. This avoids importing a height measured separately or relying on an uncertain microscope estimate. Uri Vool, group leader at the Max Planck Institute for Chemical Physics of Solids in Dresden, said that “the exact sensor height is usually unknown” in experiments and that extracting it from the same magnetic measurement removes a major source of error.
What the experiment reconstructed
The team applied the method to Fe3−xGaTe2, a material 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.
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The joint fit estimated the sensor height as 81 nanometres. That value is the result reported for this demonstration by the University of Vienna-led team in 2026; it is not a universal operating distance for every NV experiment or every ultrathin magnet.
| Quantity | How it was obtained | Reported result or qualification |
|---|---|---|
| External magnetic field | Measured with a diamond NV-center sensor | Experimental input |
| Internal magnetization pattern | Inferred by matching predicted fields while minimizing magnetic energy | Reconstructed, not directly measured point by point |
| Sensor-to-sample distance | Optimized simultaneously with magnetization | 81 nanometres in the Fe3−xGaTe2 demonstration |
| Material | Sample synthesized at the National University of Singapore | Fe3−xGaTe2 |
| Measurement location | NV magnetic sensing at the Max Planck Institute for Chemical Physics of Solids in Dresden | Specific to the reported experiment |
Why estimating height changes the result
Field strength and spatial detail both depend on how far the sensor is from the magnetic layer. If software assumes a sensor is closer than it really is, it can compensate by assigning the sample an incorrectly weak or diffuse magnetization. If it assumes the sensor is farther away, it can produce the opposite distortion. Because the decay is rapid at these length scales, even a modest height error can propagate through the entire inversion.
Jointly fitting height and magnetization makes the reconstruction internally consistent with one field map. It does not eliminate every experimental uncertainty: noise, calibration, imperfect knowledge of sample geometry and limitations in the magnetic model can still affect the result. The reported work establishes the height estimate for its Fe3−xGaTe2 data, not a guarantee that the method will recover an exact distance in all samples.
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How this differs from a conventional inversion or black-box AI
A conventional inversion may require the sensor height to be known in advance and can return one of many mathematically acceptable magnetization maps. A black-box neural network can learn visual correlations but may produce a pattern without a clear physical reason. The Vienna workflow uses the measured field, a forward magnetic model and an energy criterion together.
| Question | Physics-guided workflow reported here | What the report establishes about alternatives |
|---|---|---|
| Does it use physical constraints? | Yes. Candidate states must match the field and are ranked using magnetic energy. | Not stated for any particular competing implementation. |
| Must sensor height be supplied? | No. Height is fitted with magnetization. | Conventional approaches often require a known height, but no universal rule applies to every method. |
| Why was a solution selected? | It fits the measurement and is energetically plausible. | A black-box model’s selection rule depends on its training and architecture. |
| Numerical head-to-head accuracy benchmark | Not provided in the reported source. | Not available for a fair comparison from the reported information. |
Software availability
The workflow uses NeuralMag, an existing open-source micromagnetic package developed in Vienna. The reconstruction implementation is also described as freely available. That combination is intended to let other laboratories reproduce the optimization, adapt the physical model and apply it to their own magnetic-field measurements.
“Our hope is that other laboratories will pick up the framework and apply it to their own measurements,” Setescak said. Users should still check the project’s current documentation and license before integrating the code into a production pipeline, because the available features and installation requirements can change.
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What the method can and cannot tell you
It can provide
- An internally consistent estimate of magnetization from an external-field map.
- A sensor-height estimate derived from the same measurement rather than a separate distance assumption.
- A solution constrained by magnetic energy, reducing the set of mathematically possible but physically unlikely patterns.
- A software path that can be inspected and reused through NeuralMag and the freely available reconstruction code.
It does not provide automatically
- A direct, point-by-point observation of every magnetic moment.
- A guarantee that the lowest-energy model is the only possible state when the material is trapped in a metastable or driven configuration.
- A universal accuracy figure for different materials, sensor types or sample geometries.
- A published numerical advantage over every conventional inversion or neural-network method; the reported source does not include such a head-to-head benchmark.
Next step: three-dimensional magnetic imaging
The team is preparing an extension to three-dimensional X-ray imaging of complex magnetic nanostructures. X-ray measurements provide a different experimental modality and can involve structures that are difficult to represent with a simple ultrathin-film model. Extending the same physics-guided idea will require matching the forward model and energy terms to the geometry, contrast mechanism and constraints of 3D X-ray data.
The broader significance is methodological: when measurements underdetermine the internal state, combining data fidelity with known physical laws can turn an unstable inversion into a testable, interpretable estimate. The 81-nanometre result demonstrates that the sensor geometry itself can be inferred as part of that calculation.
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