Hard magnet tutorial – with AI-based property predictions

Introduction

• This notebook demonstrates how to use the mammos-ai package to predict hard magnet properties using pre-trained machine learning models.

• This is the same workflow as the hard magnet tutorial but uses AI models to predict hysteresis properties instead of micromagnetic simulations.

Requirements:

• Software: mammos

• Basic understanding of hard magnet tutorial

%config InlineBackend.figure_format = "retina"


import mammos_ai
import mammos_analysis
import mammos_dft
import mammos_entity as me
import mammos_spindynamics
import mammos_units as u
import matplotlib.pyplot as plt
import numpy as np

# Allow convenient conversions between A/m and T
u.set_enabled_equivalencies(u.magnetic_flux_field());

Load data from DFT and spindynamics databases

material = "Co2Fe2H4"

results_dft = mammos_dft.db.get_micromagnetic_properties(material)
results_spindynamics = mammos_spindynamics.db.get_spontaneous_magnetization(material)

Calculate micromagnetic intrinsic properties using Kuz’min formula

results_kuzmin = mammos_analysis.kuzmin_properties(
    T=results_spindynamics.T,
    Ms=results_spindynamics.Ms,
    K1_0=results_dft.Ku_0,
)

The Tc is

results_kuzmin.Tc

CurieTemperature(value=1061.566766957005, unit=K)

results_kuzmin.plot();

AI-based prediction of extrinsic properties

Rather than running micromagnetic simulations to compute hysteresis loops and derive extrinsic properties such as coercive field Hc and maximum energy product BHmax, we use pre-trained machine learning models from the mammos-ai package.

Details about the model and its training data can be found in the training repository.

This approach allows for rapid predictions of hysteresis properties based on the intrinsic micromagnetic parameters computed earlier.

The model was trained using a 50 × 50 × 50 nm cubic grain with uniaxial anisotropy, where both the anisotropy axis and the applied magnetic field were oriented along a cube edge.

T = np.linspace(0, 1.1 * results_kuzmin.Tc.q, 20)

results_ai = mammos_ai.Hc_Mr_BHmax_from_Ms_A_K(
    Ms=results_kuzmin.Ms(T),
    A=results_kuzmin.A(T),
    K1=results_kuzmin.K1(T),
)

This has returned an ExtrinsicProperties object containing the predicted hysteresis properties. This is the same object type as returned by running micromagnetic simulations and them processing them with mammos_analysis.hysteresis.extrinsic_properties in the hard magnet tutorial.

results_ai

ExtrinsicProperties(Hc=Entity(ontology_label='CoercivityHcExternal', value=array([2942256.2  , 2857558.5  , 2740996.8  , 2543269.5  , 2428668.5  ,
       2208944.   , 2115859.   , 1819307.6  , 1808106.2  , 1698485.6  ,
       1829058.5  , 1429938.1  , 1091347.4  ,  903446.3  , 1116855.1  ,
       1057900.2  ,  966576.75 ,   17457.65 ,   17204.068,   17204.068],
      dtype=float32), unit='A / m'), Mr=Entity(ontology_label='Remanence', value=array([1142329.1  , 1137940.8  , 1132234.9  , 1115691.5  , 1108781.   ,
       1087777.1  , 1069930.6  , 1039830.7  ,  997795.75 ,  947130.56 ,
        841426.94 ,  782085.25 ,  755020.2  ,  736731.56 ,  618404.6  ,
        582873.56 ,  390627.44 ,   61490.266,   57923.465,   57923.465],
      dtype=float32), unit='A / m'), BHmax=Entity(ontology_label='MaximumEnergyProduct', value=array([409878.44, 406728.28, 402647.75, 390956.53, 386122.5 , 371605.25,
       359498.5 , 339515.28, 312623.38, 281647.78, 222328.2 , 192068.8 ,
       178941.03, 170193.77, 119876.34, 106301.82,  47868.95, 349298.25,
       350828.28, 350828.28], dtype=float32), unit='J / m3'))

We can visualize the predicted extrinsic properties as a function of temperature:

Starting with the remanence Mr:

plt.plot(T, results_ai.Mr.value, linestyle="-", marker="o")
plt.xlabel(me.T().axis_label)
plt.ylabel(me.Mr().axis_label);

We known that the remanence Mr for hard magnets is approximately the saturation magnetization Ms so we expect a similar temperature dependence. When we plot them together can see that this is indeed the case however the AI model is not perfect, as Ms should be a hard upper limit for Mr and we see some predicted Mr values slightly exceeding Ms.

At higher temperatures, the predictions become less accurate. This is because some samples are no longer classified as hard magnets based on their intrinsic properties, so the model switches to a soft-magnet predictor, which is less accurate. Even more than that, the model may be extrapolating beyond its training data, leading to unphysical results.

plt.plot(T, results_ai.Mr.value, linestyle="-", marker="o", label="Mr from AI prediction")
plt.plot(T, results_kuzmin.Ms(T).value, linestyle="--", marker="x", label="Ms from Kuzmin model")
plt.xlabel(me.T().axis_label)
plt.ylabel(me.M().axis_label)
plt.legend();

We can also look how the coercive field changes with temperature:

plt.plot(T, results_ai.Hc.value, linestyle="-", marker="o")
plt.xlabel(me.T().axis_label)
plt.ylabel(me.Hc().axis_label);

Finally we look at the maximum energy product BHmax. This shows a clear error at higher temperatures, where the values jump significantly. This is again due to the model switching from hard-magnet to soft-magnet predictions as the intrinsic properties change with temperature. The input to the function at these high temperature points is outside the training data range, so the model is extrapolating and producing unphysical results.

plt.plot(T, results_ai.BHmax.value, linestyle="-", marker="o")
plt.xlabel(me.T().axis_label)
plt.ylabel(me.BHmax().axis_label);

We can check the metadata of the model to see the range of input parameters it was trained on:

mammos_ai.Hc_Mr_BHmax_from_Ms_A_K_metadata()

{'model_name': 'cube50_singlegrain_random_forest_v0.1',
 'description': 'Random forest model trained on simulated data for single grain cubic particles with 50 nm edge length with the external field applied parallel to the anisotropy axis.',
 'training_data_range': {'Ms': (Entity(ontology_label='SpontaneousMagnetization', value=79580.0, unit='A / m'),
   Entity(ontology_label='SpontaneousMagnetization', value=3980000.0, unit='A / m')),
  'A': (Entity(ontology_label='ExchangeStiffnessConstant', value=1e-13, unit='J / m'),
   Entity(ontology_label='ExchangeStiffnessConstant', value=1e-11, unit='J / m')),
  'K': (Entity(ontology_label='UniaxialAnisotropyConstant', value=10000.0, unit='J / m3'),
   Entity(ontology_label='UniaxialAnisotropyConstant', value=10000000.0, unit='J / m3'))},
 'input_parameters': ['Ms (A/m)', 'A (J/m)', 'K1 (J/m^3)'],
 'output_parameters': ['Hc (A/m)', 'Mr (A/m)', 'BHmax (J/m^3)'],
 'source': 'https://github.com/MaMMoS-project/ML-models/tree/main/beyond-stoner-wohlfarth/single-grain-easy-axis-model'}

Internally, the model checks if the sample is a hard magnet based on the intrinsic properties. If it is, it uses a hard-magnet predictor. If it is not, it switches to a soft-magnet predictor. We can directly check which samples are classified as hard magnets:

mammos_ai.is_hard_magnet_from_Ms_A_K(
    Ms=results_kuzmin.Ms(T),
    A=results_kuzmin.A(T),
    K1=results_kuzmin.K1(T),
)

array([ True,  True,  True,  True,  True,  True,  True,  True,  True,
        True,  True,  True,  True,  True,  True,  True,  True, False,
       False, False])

In this notebook, we have shown how the mammos-ai package can serve as a fast alternative to full micromagnetic simulations for predicting the hysteresis properties of hard magnets. By using intrinsic micromagnetic parameters as inputs, the model provides rapid estimates that avoid the hours-long runtime of traditional simulations. However, this speed comes with reduced accuracy, particularly when extrapolating beyond the range of the training data. Users should therefore take care when interpreting results.