Beyond Stoner-Wohlfarth: Machine-Learning Models and Symbolic Regression of Hard-Magnet Properties

Technical Summary: Beyond Stoner–Wohlfarth: Machine-Learning Models and Symbolic Regression of Hard-Magnet Properties

Problem Statement
Predicting the extrinsic magnetic properties of permanent magnets—specifically the coercive field ($H_c$), remanent magnetisation ($M_r$), and maximum energy product ($BH_{max}$)—from their intrinsic micromagnetic parameters (saturation magnetisation $M_s$, exchange constant $A$, and uniaxial anisotropy constant $K$) is a fundamental challenge in materials science. While established analytical models (e.g., Stoner–Wohlfarth and Kronmüller) provide useful estimates, they often fail to account for non-uniform magnetisation processes and microstructural complexities. Conversely, direct micromagnetic simulations, which capture these details, are computationally expensive, often requiring hours or days per parameter set, thereby hindering rapid material exploration and optimization.

Methodology
The authors address this gap by training machine learning (ML) surrogate models on a dataset of 12,012 micromagnetic simulations of an idealised isolated cubic grain (50 nm side length). The simulations were performed using the MaMMoS-MuMag code (a JAX-based finite-element solver) across broad ranges of intrinsic parameters:

• $M_s$: $7.96 \times 10^4$ to $3.98 \times 10^6$ A/m
• $A$: $10^{-13}$ to $10^{-11}$ J/m
• $K$: $10^4$ to $10^7$ J/m³

The dataset was filtered to ensure physical validity (e.g., excluding cases where the exchange length or domain-wall width was smaller than the 1 nm mesh size) and to remove simulations where magnetisation did not reverse within the applied field range. This resulted in 9,957 valid samples for training.

The study employs a multi-stage approach:

1. Classification: A k-means clustering algorithm distinguishes between "hard" and "soft" magnet regimes based on the normalised remanence ($M_r/M_s$).
2. Forward Modeling: Five ML architectures (Linear Regression, LASSO, Random Forest, Gaussian Process, and Fully Connected Neural Network) were trained to predict $H_c$, $M_r$, and $BH_{max}$ from $M_s$, $A$, and $K$.
3. Symbolic Regression: Using the PySR library, the authors searched for closed-form analytical expressions that fit the simulation data, aiming to recover interpretable physical laws.
4. Inverse Modeling: The mapping was inverted to predict intrinsic parameters ($M_s$, $A$, $K$) from the extrinsic outputs ($H_c$, $M_r$, $BH_{max}$).

Key Contributions and Results

• Superior Predictive Accuracy: Benchmarked against analytical models on a held-out test set, the ML models significantly outperformed traditional analytical approaches. The Random Forest (RF) and Gaussian Process (GP) models achieved the lowest errors. For the coercive field, the RF reduced the Root Mean Square Error (RMSE) from 195 kA/m (fitted Kronmüller model) to 92 kA/m, while the GP achieved 18 kA/m. Similar improvements were observed for $M_r$ and $BH_{max}$.
• Symbolic Regression Discoveries: The symbolic regression successfully recovered the Kronmüller form for the coercive field but with a novel twist: the effective demagnetising factor ($N_{eff}$) was found to be a function of material properties rather than a constant. Specifically, the derived law is $H_c = [\alpha - n \ln \tilde{L}/\kappa] H_A$, where $\tilde{L}$ is the reduced grain size and $\kappa$ is the magnetic hardness parameter. The authors also discovered new closed-form expressions for $M_r$ and $BH_{max}$ that include corrections based on grain size and anisotropy, which are not present in standard literature. These symbolic laws, containing at most two fitted constants, approached the accuracy of the ML surrogates.
• Feature Importance and Inverse Problem: Analysis revealed that $M_s$ and $K$ are the dominant factors determining extrinsic properties, while the exchange constant $A$ has a weak influence. Consequently, the inverse problem (recovering intrinsic parameters from extrinsic ones) yielded accurate results for $M_s$ and $K$ but failed to accurately recover $A$. The authors attribute this not to model failure, but to the physical reality that $A$ influences the hysteresis loop only weakly, making it impossible to uniquely constrain from the three extrinsic metrics alone.
• Computational Efficiency: The trained pipeline, released via the mammos-ai Python package, enables the screening of thousands of candidate parameter sets in seconds. This represents a speed-up of over three orders of magnitude compared to direct micromagnetic simulations (which take ~147 minutes per sample on a GPU).

Significance and Scope
The paper claims to provide a robust, data-driven framework for bridging the gap between intrinsic material parameters and extrinsic magnetic performance. By combining high-accuracy ML surrogates with interpretable symbolic regression, the work offers both predictive power and physical insight, recovering established analytical forms while identifying necessary corrections.

The authors explicitly limit the scope of their claims to an idealised, isolated cubic grain with the field aligned to the easy axis. They acknowledge that real materials involve microstructural defects, grain boundaries, misorientation, and thermal activation, which are absent in their model. Therefore, the models predict the properties of an ideal defect-free grain; discrepancies with experimental measurements in real magnets are attributed to these missing microstructural factors. The primary utility of the work is the rapid screening of candidate parameter sets to guide further, more detailed investigations or to understand the fundamental limits of ideal grain performance. The models and laws are made available to the community to facilitate this rapid exploration.