Hysteresis modeling

Requirements

Data

We acquire for each magnet:

• Measured current $I_{\text{m}}$
• Programmed current $I_{\text{P}}$
• Measured field $B$
• Measured field derivative $\dot{B}$

at a fixed sampling rate of 1 kHz

Since 2025 we in addition have the measured voltage of the coil of the magnet, $V$.

The experimental data that we acquire can then be expressed as $\mathcal{D}=\{I_{\text{m},t},I_{\text{p},t},B_t,{\dot{B}}_t{\}}_{t=1,\dots ,T}$

We can use either the measured, or programmed $I$ for the modeling. The programmed is preferred.

Model

As we are aiming to resolve small errors in our data that arise due to hysteresis, we construct the following model:

$$B_t=h(I_{t)}+\bar{f}(I_t)+\epsilon$$

where $h(I_t)$ is a polynomial fit of the data using least squares fitting. $\bar{f}(I_t)$ is a model that we fit using one of our approaches.

The approaches for fitting $\bar{f}(I_t)$ are the following:

• Transformers for Time Series
  • Standard transformers (fixed context length)
  • Decoder-only transformers
  • “Recurrent” transformers like Transformer-XL
• Physics-based neural networks
  • PhyLSTM to solve a differential equation like Bouc-Wen model
  • Preisach Neural Networks
• Deterministic physics approaches
  • Like solving the Preisach model

Modeling approaches

We can model $\{I,\dot{I},B_{\text{past}}\}\to B_{\text{future}}$ with an encoder-decoder style prediction scheme.

An alternative is to model $\{I,\dot{I},{\dot{B}}_{\text{past}}\}\to {\dot{B}}_{\text{future}}$ by either differentiating the measured $B$, or computing the $\Delta B$, or using the measured $\dot{B}$ directly, and then integrating the $\dot{B}$ to retrieve $B$, however this requires an integration constant to start from.

Results

	(‘validation RMSE ($10^{-4}T)$’, ‘non-cycled’)	(‘validation RMSE ($10^{-4}T)$’, ‘cycled’)	(‘validation RMSE ($10^{-4}T)$’, ‘AR’)	(‘test RMSE ($10^{-4}T)$’, ‘non-cycled’)	(‘test RMSE ($10^{-4}T)$’, ‘cycled’)	(‘test RMSE ($10^{-4}T)$’, ‘AR’)
EDLSTM	0.356296	0.497344	0.433432	1.4794	1.54061	6.00721
ATTNLSTM	0.415864	0.442925	0.585709	2.36485	1.53852	5.73712
TFLSTM	0.265873	0.354366	0.368044	1.09261	1.28251	5.27817
TFT	0.422739	0.655904	0.987159	1.52829	1.32077	5.80831
TFT	0.176099	0.422175	0.634103	0.332851	0.430202	0.457526