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

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Machine-learning surrogates and symbolic regression can replace expensive micromagnetic simulations for predicting hard-magnet hysteresis properties, cutting held-out coercive-field error from 195 to 18 kA/m while recovering the Kronmüller

desk verdict A useful, honest surrogate-modeling paper for ideal hard-magnet grains; the ML gains are real, the symbolic laws are partly rediscoveries with an unvalidated size ansatz, and the headline errors apply only to a target-conditioned subset. read the letter →

arxiv 2607.29249 v1 pith:7T6JAK6D submitted 2026-07-31 cond-mat.str-el cond-mat.mtrl-sci

Samuel J. R. Holt , Christina Winkler , Timoteo Colnaghi , Martin Lang , Swapneel A. Pathak , Andrea Petrocchi , Michael Adams , Thomas Schrefl
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Andreas Marek Hans Fangohr
This is my paper · ORCID
classification cond-mat.str-elcond-mat.mtrl-sci
keywords micromagneticsimulationhysteresismodellinghardmagnetscoercivefieldsymbolicregressionmachine-learningsurrogateinversedesignpermanent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the expensive step in permanent-magnet modelling—computing hysteresis properties from micromagnetic parameters—can be replaced by fast surrogate models and, more strongly, by compact closed-form laws. Using roughly twelve thousand micromagnetic simulations of an idealized 50-nm cube as ground truth, it shows that random-forest and Gaussian-process surrogates predict coercive field, remanence, and maximum energy product with substantially lower held-out error than the standard analytical models. Symbolic regression then recovers the Kronmüller form of coercivity, with an effective demagnetising factor that depends on the material, and produces new closed-form expressions for remanence and energy product with at most two fitted constants each. The practical payoff is that thousands of candidate parameter sets can be screened in seconds rather than hours. The paper is careful to state that these results are conditional on the hard-magnet subset and the 50-nm geometry.

What carries the argument

The analysis is built on two dimensionless combinations of the intrinsic parameters: the hardness parameter κ = sqrt(K / (µ0 Ms²)) and the reduced grain size L̃ = L / ℓex, with ℓex the exchange length. Scaling each extrinsic target by its natural analytical unit (the anisotropy field HA for Hc, Ms for Mr, µ0Ms²/4 for BHmax) lets symbolic regression search a compact dimensionless space; the key output is the effective demagnetising factor Neff = 2n κ ln L̃, which converts the fitted constant in the Kronmüller coercivity formula into a material-dependent quantity. The same construction yields correction terms in the Mr and BHmax laws that grow when the grain spans many exchange lengths.

What would settle it

Run the same symbolic-regression search on the full unfiltered population, or on cubes of side 100 nm and 200 nm with identical sampling; if the fitted exponents of L̃ in Eqs. (10)–(11) shift beyond the reported standard errors, the recovered laws are artefacts of the 50-nm conditioning rather than general relations.

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Extended reading notes

Core claim

The paper establishes that, for an idealised 50-nm cubic hard-magnet grain, the mapping from the three intrinsic constants (saturation magnetisation Ms, exchange stiffness A, uniaxial anisotropy K) to the three extrinsic hysteresis properties (coercive field Hc, remanence Mr, maximum energy product BHmax) is learnable and compressible. Trained on 12,012 micromagnetic simulations, random-forest and Gaussian-process surrogates predict all three extrinsic quantities on held-out data with substantially lower error than the best analytical benchmarks: Hc RMSE of 92 and 18 kA/m versus 195 kA/m for the fitted Kronmüller form, Mr RMSE of 11 and 4 versus 53 kA/m, and BHmax RMSE of 17 and 2 versus 148

Load-bearing premise

The load-bearing premise is that the reported accuracies and formulas describe hard-magnet grains in general; in fact they are conditional on the paper's own filtering (reversal within −10 T, Hc and Mr above 10⁴ A/m, length scales above the 1-nm mesh, and a k-means hard/soft split) and on a single 50-nm cube with easy-axis field at zero temperature, a point the paper itself flags in Sections IV A and IV E.

Editorial extensions

If this is right

  • A material scientist can evaluate a candidate (Ms, A, K) triple in about a second per material and batch-screen 10,000 candidates in under three seconds, compared with a median 147 minutes per micromagnetic simulation.
  • For the four published hard-magnet parameter sets tested, both the released pipeline and the symbolic laws reproduce dedicated simulations to within 12% for coercivity, 5% for energy product, and 3% for remanence.
  • The recovered coercivity law is the Kronmüller form with Neff = 2nκ ln L̃, so the old fitted constant becomes a material-dependent quantity.
  • The inverse models recover Ms and K from Hc, Mr, and BHmax accurately, but A cannot be recovered from these three outputs, making the inverse problem underdetermined for exchange stiffness.
  • The analytical hardness criterion κ > 1/√6 classifies 96.3% of simulated grains as hard or soft, with errors only in the conservative direction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same dimensionless construction is applied to other grain shapes, the correction structure may carry over, giving a template for shape-agnostic hard-magnet surrogates.
  • The near-zero influence of A on the extrinsic properties implies that inverse design from hysteresis alone is fundamentally underdetermined; a complementary measurement that couples to exchange, such as ferromagnetic-resonance frequency or nucleation-field angular dependence, would be needed to pin A.
  • The paper's own caveat that the size dependence is an ansatz suggests a direct multi-size test: if the fitted exponents of L̃ change across cube sizes, the formulas are geometry-conditioned fits rather than general laws.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper trains random-forest, Gaussian-process, and other ML surrogates on ~12,000 micromagnetic simulations of an isolated 50-nm cubic grain with uniaxial anisotropy, sampling Ms, A, and K over broad ranges. On a held-out test set restricted to a k-means-defined 'hard-magnet subset' (8,831 samples), the random forest and Gaussian process predict Hc, Mr, and BHmax with substantially lower RMSE than fitted analytical benchmarks (Kronmüller, Mr ≈ Ms, and μ0Ms²/4). Symbolic regression yields three compact closed-form laws, Eqs. (9)–(11), which the authors interpret as recovering the Kronmüller form for Hc with a material-dependent effective demagnetizing factor, and as new corrections for Mr and BHmax. The inverse problem is also studied: Ms and K are recovered well, while A is not. A released package, mammos-ai, implements the screening pipeline.

Significance. If the claims hold, this is a useful contribution to computational magnetism: it provides a reproducible, openly released ML surrogate that runs orders of magnitude faster than micromagnetics, and it proposes compact closed-form corrections that approach the accuracy of a random forest. The paper is also transparent in releasing data, code, and fixed-seed figure scripts, and it fits the Kronmüller α on training data only. The main risk, which the authors partly acknowledge, is that all headline results are conditioned on a target-derived population; this must be resolved before the symbolic laws can be regarded as general closed-form expressions.

major comments (3)
  1. [§IV A, §IV E (Eqs. 9–11)] The population on which the models are trained and evaluated is conditioned on the target variables: 1,624 non-reversing simulations (no reversal by μ0H = −10 T) are discarded, 431 samples below the Hc/Mr floors are removed, and the hard-magnet subset is defined by k-means on (Ms, Mr/Ms). Section IV A acknowledges that this filtering couples inputs to targets, but the symbolic-regression laws are nevertheless presented as 'new closed-form expressions' without testing on the excluded samples. A concrete and decisive test is to evaluate Eq. (9) on the 1,624 non-reversing samples: for these, the simulated Hc is beyond the applied-field range, so the law should predict Hc ≥ 10 T. If it does not, the recovered functional form and the reported RMSEs are artifacts of the selection. At minimum, the claims must be explicitly restricted to the target-conditioned hard-magnet population.
  2. [§IV E, Eq. (12)] The claimed recovery of the Kronmüller form is partly shaped by the dimensionless normalization Hc/HA, which is exactly the reduced variable used in the Kronmüller model. The expression Neff = 2n κ ln L̃ is a rewriting of the fitted expression rather than an independent measure, and the L-dependence is explicitly an ansatz (stated in §IV E). The paper should make this caveat more prominent and avoid implying that a validated size dependence has been discovered. In particular, Eqs. (9)–(11) apply only to the 50-nm cube geometry and to the conditioned population; their presentation as general closed-form laws is premature without additional geometry or excluded-sample checks.
  3. [§IV B] The hard/soft classification that defines the hard-magnet subset is based on k-means on (Ms, Mr/Ms)—i.e., on a target variable. While the comparison to the κ criterion in Fig. 2 is a legitimate classification test, the subsequent analysis inherits this target-derived definition. The hard-magnet subset is not the same as the set of materials satisfying the physical criterion Eq. (2): 385 hard magnets lie below the κ = 1/√6 line. The paper should clarify that the reported laws and error metrics apply to this cluster-defined subset, not to all materials that pass the analytical hardness criterion.
minor comments (4)
  1. [Table I] The table header lists 'Linear Regression (LP)' but the text and the table row use 'LR'. This should be made consistent.
  2. [Table II, FCNN row] The FCNN results for A are numerically absurd (MAPE 3.5 × 10^11 %, RMSE 7.8 × 10^9 pJ/m), which suggests a convergence or scaling problem rather than a meaningful model failure. This should be remarked on or the model excluded from the comparison.
  3. [Fig. 9 caption] The caption contains the fragment 'An invalid sample in Sec. III', which is incomplete and does not explain the figure. Please rewrite the caption to describe the pipeline flow.
  4. [§IV E] The rounding/simplification step in the symbolic-regression procedure is described only verbally. For reproducibility, the original Pareto-front expression before exponent rounding should be reported, at least in the Supplemental Material.

Circularity Check

3 steps flagged · score 5.0 of 10

Symbolic-regression 'laws' are partly constructed, not purely discovered: Eq. (11) is the square of the remanence law by the paper's own statement; the Kronmüller-form 'recovery' is partly forced by scaling the SR target by HA; and the laws are fit/evaluated on a target-selected population. The ML-versus-analytical benchmarks remain independent.

  1. self definitional [Section IV E (dimensionless formulation; Eqs. (9), (12))]
    "To make the search efficient and its results physically meaningful, we work in dimensionless form. Each extrinsic property is scaled by its natural analytical unit: the coercive field by the anisotropy field of Eq. (5) ... Multiplying out Eq. (9) gives Hc = αHA − ( 2n κln ˜L)Ms, which is exactly the Kronmüller form of Eq. (6), in which the effective demagnetising factor is no longer a fitted constant but a function of the material, Neff = 2n κln ˜L."

    The SR target is the ratio Hc/HA, so any fitted expression f(κ, L̃) automatically yields a law Hc = f(κ, L̃)·HA; the HA proportionality of the 'recovered' Kronmüller αHA term is imposed by the chosen normalisation, not found by the search. Since HA = 2κ²Ms, every such expression is identically rewritable as αHA − Neff Ms with Neff = (α − f)HA/Ms, so the two-term Kronmüller structure and the 'material-dependent demagnetising factor' Neff = 2n κ ln L̃ are reparametrisations of the same fit, not independent measurements. The genuinely empirical part is the specific ln L̃/κ dependence (and the L-dependence is itself an admitted ansatz).

  2. renaming known result [Section IV E, Eqs. (10) and (11); Abstract]
    "Equation (11), in turn, is the square of the remanence law scaled by µ0M 2 s /4: the search independently rediscovers the energy-product relation of Eq. (8), with the remanence correction folded in. This is to be expected for a square loop."

    By the paper's own sentence, Eq. (11) — BHmax = (1 − εb L̃/κ⁴)² µ0Ms²/4 — is the square of the Eq. (10) remanence expression (with one refitted constant) times the fixed analytical unit µ0Ms²/4, i.e., the benchmark relation Eq. (8) evaluated with the fitted remanence. The abstract's claim of 'new closed-form expressions for the remanence and maximum energy product' therefore presents a derived corollary as an independent law: the three headlined laws reduce to two plus an input benchmark. Because BHmax was extracted using a demagnetising coefficient of 1/3 and Eq. (8) is one of the analytical benchmarks, the square-loop structure is also partly encoded in the target extraction.

1 more flagged steps
  1. fitted input called prediction [Section III (data preparation); Section IV A; Section IV C 2; Section IV E]
    "If the magnetisation did not reverse by µ0H = −10 T the corresponding outputs were recorded as invalid. ... Samples with Mr < 10⁴ A/m or Hc < 10⁴ A/m are excluded before training. ... The clustering separates the valid simulations into 8831 hard magnet cases and 1557 soft magnet cases. ... Strong agreement is expected here, because the hard-magnet subset was itself selected on Mr/Ms. The comparison is therefore a consistency check rather than an independent prediction."

    Eqs. (9)–(11) and the Table I RMSEs are fitted and held-out-evaluated on a population selected by the target variables: 1,624 non-reversing simulations (Hc beyond the −10 T sweep ceiling) are discarded, Mr/Hc floors are imposed, and the 'hard-magnet subset' is defined by k-means on (Ms, Mr/Ms). The paper concedes for Mr ≈ Ms that agreement is a 'consistency check rather than an independent prediction'; Eq. (10) inherits that conditioning — its εm correction is fitted inside a population with Mr/Ms near 1 by construction. The Hc law is likewise fit on an Hc-truncated distribution, so the recovered form may absorb the selection (input–target coupling admitted in IV A). Held-out error within this conditioned population does not establish the laws over the stated broad ranges.

full rationale

What is not circular: the central ML benchmark is self-contained and reproducible — five surrogates are compared with the Stoner–Wohlfarth, fitted Kronmüller, Mr = Ms, and µ0Mr²/4 benchmarks on an identical fixed-seed held-out 20% test set (Table I, Figs. 3–6), with code, data, and weights released (Refs. [9, 10, 27, 37, 38]). A surrogate that beats analytical benchmarks on held-out micromagnetically simulated data is independent content; the self-citations here are data/software availability, not load-bearing arguments, and no uniqueness theorem or ansatz is imported through an author self-citation chain. What is partial: (1) the 'recovery of the Kronmüller form' is weaker than it appears because the SR target is Hc/HA and the inputs are κ and L̃; every fitted f(κ,L̃) yields Hc = f·HA, and via HA = 2κ²Ms any such expression is identically a two-term Kronmüller form, so 'Neff = 2n κ ln L̃' is an algebraic renaming of the fit (the specific ln L̃/κ dependence is nonetheless a genuine empirical finding). (2) Eq. (11) is, by the paper's own words, the square of Eq. (10) times µ0Ms²/4 — the 'new' BHmax law is a corollary of the Mr law plus the benchmark Eq. (8). (3) The laws and the headline errors are produced by, and validated on, a target-conditioned population (non-reversal exclusions, Mr/Hc floors, k-means on (Ms, Mr/Ms)); the paper's own remark that Mr ≈ Ms agreement is 'a consistency check rather than an independent prediction' applies equally to the fitted remanence law. The manuscript itself flags its constraints — Section IV A admits the filtering couples inputs to targets, and Section IV E states the L-dependence 'is an ansatz... rather than a validated size dependence' — so the circularity is disclosed and partial, not concealed. The ML benchmarking and the specific functional-form search retain independent content, so the score sits in the 4–6 band; I assign 5 because more than one headlined result is partly constructed while the core empirical comparisons stand.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claims rest on fitted constants and a conditioned dataset rather than on first-principles derivation: the symbolic laws contain four fitted constants plus hand-rounded exponents, and the benchmark uses the same solver that generated the data. No new physical entities are postulated. Independence claims hang on the held-out 20% of a target-conditioned dataset.

free parameters (6)
  • α (Hc law leading constant) = 0.942 ± 0.004
    Fitted on the training hard-subset by least squares; leading constant in Eq. (9); strongly correlated with n (ρ = 0.96).
  • n (Hc law log-slope) = 0.0921 ± 0.0004
    Fitted; sets the material-dependent demagnetising factor Neff = 2n κ ln L̃ in the rewritten form Eq. (12).
  • εm (Mr correction amplitude) = (5.18 ± 0.02) × 10⁻⁵
    Fitted; amplitude of the remanence correction in Eq. (10).
  • εb (BHmax correction amplitude) = (7.81 ± 0.03) × 10⁻⁵
    Fitted; amplitude in Eq. (11), which the paper notes equals the square of the Mr correction scaled by µ0Ms²/4.
  • Rounded exponents (L̃¹, κ⁴, factor 2) = 1, 4, 2
    Chosen by rounding near-rational values 'wherever this does not degrade the fit on the training set' (Section IV E); hand-selected, not independently validated, and the L-dependence is explicitly an ansatz.
  • Kronmüller benchmark α = 0.849
    Fitted on the training set for the analytical benchmark of Eq. (6) with Neff = 1/3 (Section IV C 1).
assumptions (5)
  • domain assumption The micromagnetic solver (MaMMoS-MuMag, finite-element, quasistatic energy minimisation) produces exact ground truth for Hc, Mr, and BHmax.
    All ML training and all benchmarks treat solver output as exact; discretisation errors from the single reused 1-nm mesh are not propagated into reported errors (Section III).
  • domain assumption Zero-temperature quasistatic reversal with a ~1° field misalignment represents the ideal-grain hard-magnet regime.
    No thermal activation is included; real magnets reverse at lower fields; acknowledged in Section V.
  • ad hoc to paper The k-means partition on (Ms, Mr/Ms) correctly defines the hard/soft regime of interest.
    Labels are derived from simulation outputs and set the evaluation population for every headline number (Section IV B).
  • standard math For fixed grain geometry, {κ, L̃} is the complete dimensionless input set.
    Dimensional analysis with fixed L = 50 nm (Section IV E); sound but assumes cube edge and demagnetising factor are fixed.
  • domain assumption The sphere-derived hardness criterion κ > 1/√6 is a meaningful benchmark for cubes.
    Used in Eq. (2) and tested on the filtered valid set only, which excludes no-reversal (ultra-hard) samples (Section IV B).

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Pith. "Pith review of Beyond Stoner-Wohlfarth: Machine-Learning Models and Symbolic Regression of Hard-Magnet Properties." pith.science (2026). https://pith.science/paper/7T6JAK6D

@misc{pith2026260729249,
  author       = {Pith},
  title        = {Pith review of: Beyond Stoner-Wohlfarth: Machine-Learning Models and Symbolic Regression of Hard-Magnet Properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7T6JAK6D}},
  note         = {Machine review of arXiv:2607.29249}
}
read the original abstract

Predicting the extrinsic properties from hysteresis loops of a magnetic grain, namely the coercive field, remanent magnetisation, and maximum energy product, from its intrinsic micromagnetic parameters is a central problem in permanent-magnet modelling. Established analytical models provide useful estimates but often neglect nonuniform magnetisation processes, whereas direct micromagnetic simulations are computationally expensive. In this work, we train machine-learning models on 12012 micromagnetic simulations of an idealised cubic grain, spanning broad ranges of the saturation magnetisation, exchange constant, and uniaxial anisotropy constant. Benchmarked against the analytical models on identical held-out data, the machine-learning models predict all three extrinsic properties with substantially lower errors. Symbolic regression recovers the Kronm\"uller form of the coercive field, with an effective demagnetising factor that depends on the material, and finds new closed-form expressions for the remanence and maximum energy product. Each law contains at most two fitted constants yet approaches the accuracy of the machine-learning models. We also investigate the inverse problem of recovering the intrinsic parameters from the three extrinsic properties. The saturation magnetisation and anisotropy constant are recovered accurately, whereas the exchange constant is not, because it influences the extrinsic properties only weakly. The trained models are released through the mammos-ai Python package, enabling thousands of candidate parameter sets to be screened in seconds rather than the hours or days required by direct micromagnetic simulation.

Figures

Figures reproduced from arXiv: 2607.29249 by the authors.

Figure 1
Figure 1. FIG. 1. Pearson (left) and Spearman (right) correlation coefficients between intrinsic and extrinsic properties. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. shows the normalised remanence of every valid simulation against the hardness parameter κ. The cri￾terion, derived for a uniformly magnetised sphere, de￾scribes our cubic grains well: it classifies 96.3% of the samples correctly. Its errors are entirely one-sided. No soft magnet is ever classified as hard, while 385 simula￾tions remain hard below the κ = 1/ √ 6 boundary; for the hard class, this corresponds to a pre… view at source ↗
Figure 4
Figure 4. FIG. 4. Analytical estimate of the remanent magnetisation [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (6 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Analytical estimates of the coercive field for the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Analytical estimate of the maximum energy product [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Random forest predictions of the extrinsic properties against the simulated values for the hard-magnet subset. The test [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Symbolic-regression predictions of the extrinsic properties against the simulated values for the hard-magnet subset, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Random forest predictions of the intrinsic parameters from the extrinsic properties, against the true values for the [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Overview of the ML model pipeline. An invalid [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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    We first evaluate the analytical coercivity models introduced above on the hard-magnet subset

    Coercive field Hc Initially we will consider the forward problem: pre- dicting the extrinsic properties Hc, Mr, and BHmax from the intrinsic parameters Ms, A, and K of a hard mag- net. We first evaluate the analytical coercivity models introduced above on the hard-magnet subset. As shown in Fig. 3 (top), the Stoner–Wohlfarth estimate of Eq. (5) 5 0 2 4 6 ...

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    Maximum energy product BHmax We next evaluate the analytical estimate for the max- imum energy product on the hard-magnet subset. As shown in Fig. 5, the analytical expression of Eq. (8) cap- tures the simulated BHmax values well, with a test RMSE of 148 kJ/m3 (Table I). It remains a systematic upper bound: it assumes an ideal square demagnetisation curve...

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    Results Table I reports the errors of all five models on the held-out test set, together with the analytical estimates of Section IV C evaluated on the identical test samples and the symbolic-regression laws derived in Section IV E. The random forest (RF) and the Gaussian process (GP) provide the best predictive performance. For the coercive field, the fi...

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    Feature Importance The feature importance of the forward RF model is shown in the Supplemental Material. It is quantified by the mean decrease in impurity: each node of a regres- sion tree splits on the input variable that most reduces the variance of the target values within that node, and a variable’s importance is the total variance reduction it produc...

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    Results The three intrinsic parameters are not recovered equally well. Ms and K are recovered accurately, with test RMSEs of 14 kA /m and 100 kJ /m3, as shown in Figs. 8(a,b) and (e,f) and Table II. The exchange con- stant A is not: its test RMSE of 2 .1 pJ/m is more than a fifth of the sampled range of 0 .1–10 pJ/m, and the pre- dictions in Fig. 8(c,d) s...

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Reviewed August 3, 2026 · model on record in the stance chip above.