Tangent-Plane Evidential Uncertainty in Active Learning for Magnetic Interatomic Potentials

Abstract: Magnetic interatomic potentials need to account for coupled lattice and spin degrees of freedom, yet constructing reliable training sets remains costly because noncollinear first-principles labels are expensive. Active learning can mitigate this cost, provided that the uncertainty estimate is physically meaningful for the magnetic-response targets that drive spin reorientation. Here we extend the $\mathrm{e}<sup>2\mathrm{IP}$ evidential framework to magnetic machine-learning interatomic potentials by formulating the projected spin-force likelihood and the corresponding epistemic uncertainty in the tangent plane orthogonal to the local spin direction. This construction prevents the uncertainty model from allocating probability mass to a radial spin component that is absent from the constrained-moment supervision. Using bulk BiFeO$<em>3$ and monolayer CrTe$_2$ as benchmark systems, we show that the resulting tangent-plane epistemic uncertainty indicator $U</em>{\mathrm{epi}}<sup>{\mathrm{sf}}$ correlates strongly with prediction error and selects more informative configurations than random sampling, simultaneously improving energy, force, and projected spin-force accuracy. These results demonstrate a physically interpretable and data-efficient route for constructing uncertainty-aware magnetic machine-learning interatomic potentials.

• The paper develops a tangent-plane evidential uncertainty model for projected spin forces, achieving Spearman error correlations up to 0.919 and preserving consistency with a shared energy surface.
• Active learning using the two-dimensional epistemic uncertainty reduces BiFeO3 energy, force, and projected spin-force errors by approximately 36%, 23%, and 24%, respectively, compared with random sampling.
• The method also improves CrTe2 predictions and shows that uncertainty geometry should match the supervised physical degrees of freedom, although broader materials and acquisition strategies remain to be tested.

Motivation and problem statement

Magnetic machine-learning interatomic potentials (MLIPs) must resolve coupled lattice and spin degrees of freedom, and their training sets are expensive to build because reference labels require noncollinear or constrained-moment density-functional-theory (DFT) calculations. Active learning can reduce this cost, but only if the acquisition signal is physically meaningful for the magnetic-response targets that drive spin reorientation. This paper extends Equivariant Evidential Deep Learning for Interatomic Potentials ($\mathrm{e}^2\mathrm{IP}$) (Wang et al., 11 Feb 2026) to SOC-aware magnetic MLIPs by reformulating the evidential uncertainty geometry so that it matches the tangent-plane structure of projected spin-force learning. The central observation is geometric: in constrained-moment supervision, the learned spin-response target is not a free three-dimensional vector but the spin force $\mathbf f_{s,\perp}$ projected onto the plane orthogonal to the local spin direction. A naive three-dimensional uncertainty model would allocate probability mass to a radial component that is absent from the supervision, degrading both interpretability and error correlation.

Method

The backbone follows the SpinGNN++ decomposition, with total energy written as a sum of structural, exchange, anisotropy, biquadratic, and SOC contributions, implemented through an SOC-aware Spin-Allegro-style local-environment network (Feng et al., 2023). All response observables are energy derivatives: atomic forces from $\partial E/\partial \mathbf r$, and raw spin forces $$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$ from the derivative with respect to the local spin-direction unit vector. The supervised target is the projected quantity

$$\mathbf f_{s,\perp} = \mathbf P \mathbf f_s, \qquad \mathbf P = \mathbf I - \hat{\mathbf s}\hat{\mathbf s}^{\top},$$

consistent with the transverse character of constrained noncollinear DFT signals [(Shapeev, 2015)-style constrained-DFT references; 054420 Phys. Rev. B]. Projected spin-force errors are consequently reported in energy units (meV) rather than force units.

The uncertainty model retains the $\mathrm{e}^2\mathrm{IP}$ Normal–Inverse–Wishart evidential prior over a Gaussian likelihood, with the canonical uncertainty-shape matrix $\boldsymbol\Sigma_{0,i}=\exp(\mathbf S_i)$ built from an equivariant symmetric representation. The key modification is that this covariance is projected into the tangent subspace via $$\boldsymbol\Sigma_{0,i,\perp}=\mathbf P_i\boldsymbol\Sigma_{0,i}\mathbf P_i$$ and evaluated in a $d=2$ local orthonormal tangent basis constructed by Gram–Schmidt against a fixed Cartesian reference axis; the resulting negative log-likelihood and uncertainty score are invariant under in-plane rotations of that basis. The predictive mean is identified with the projected energy gradient rather than an independent head, so all observables remain tied to a single spin-lattice energy surface. Evidence parameters are parameterized as $\nu_i = \mathrm{softplus}(\hat\nu_i)+(d+2)$ and $\mathbf f_{s,\perp}$0, and training uses the multivariate Student-$\mathbf f_{s,\perp}$1 negative log-likelihood on tangent-plane coordinates plus an evidence regularizer. The structure-level acquisition score is the atom-averaged trace of the two-dimensional epistemic covariance,

$\mathbf f_{s,\perp}$2

Candidates are generated by spin-lattice dynamics, scored by $\mathbf f_{s,\perp}$3, labeled with DFT, and the model is retrained iteratively.

Uncertainty–error correlation

A necessary condition for $\mathbf f_{s,\perp}$4 to serve as an acquisition function is that it tracks actual prediction difficulty. For bulk BiFeO$\mathbf f_{s,\perp}$5, the Spearman correlation between $\mathbf f_{s,\perp}$6 and force RMSE is 0.919, and with projected spin-force RMSE it is 0.879; for monolayer CrTe$\mathbf f_{s,\perp}$7, the corresponding values are 0.857 and 0.848. A force-only ablation provides an informative control: the force-branch indicator $\mathbf f_{s,\perp}$8 correlates strongly with force RMSE ($\mathbf f_{s,\perp}$9 for BiFeO$\partial E/\partial \mathbf r$0, $\partial E/\partial \mathbf r$1 for CrTe$\partial E/\partial \mathbf r$2) but degrades for projected spin-force RMSE ($\partial E/\partial \mathbf r$3 and $\partial E/\partial \mathbf r$4, respectively). This directly supports the paper's geometric argument: a three-dimensional uncertainty estimator cannot fully substitute for a tangent-plane-consistent one when the target lives in the tangent space. The implication is that uncertainty geometry should be matched to the supervised degrees of freedom, not merely appended to the architecture.

Active-learning benchmarks

Two complementary systems are used: bulk BiFeO$\partial E/\partial \mathbf r$5, a room-temperature multiferroic with coupled structural and magnetic order, and monolayer CrTe$\partial E/\partial \mathbf r$6, a two-dimensional ferromagnet with strong spin-lattice sensitivity. In both cases, active learning and random sampling share the same initial 1000 configurations; active learning then adds uncertainty-selected configurations while the baseline adds random ones. Test-set results are:

System	Sampling	Energy (meV/atom)	Force (meV/Å)	Proj. spin force (meV)
BiFeO$\partial E/\partial \mathbf r$7	Random	0.286	20.0	0.579
BiFeO$\partial E/\partial \mathbf r$8	Active	0.182	15.4	0.438
CrTe$\partial E/\partial \mathbf r$9	Random	0.198	9.8	0.537
CrTe$$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$0	Active	0.174	8.0	0.451

For BiFeO$$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$1 (2000 training/validation, 878 test), active learning reduces energy MAE by roughly 36%, force MAE by 23%, and projected spin-force MAE by 24%. For CrTe$$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$2 (3000 training/validation, 800 test), improvements are smaller but consistent across all three metrics. Notably, a spin-force-targeted acquisition signal also improves atomic-force accuracy, which the authors attribute to all response observables being tied to a single learned energy surface: configurations under-constrained in their projected spin-force response correspond to regions of that surface that are also under-determined for forces. The cross-system consistency suggests the approach is not specific to one material class, though the evidence base remains limited to these two benchmarks and a single acquisition protocol.

Limitations and open questions

The paper concedes several constraints explicitly. Validation covers only two representative systems and one active-learning protocol; generality across broader magnetic materials, alternative acquisition criteria, and downstream spin-lattice simulations remains untested. The relative gains on CrTe$$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$3 are modest compared with BiFeO$$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$4, and the paper does not analyze why—whether this reflects dataset size, material dimensionality, or saturation of the random baseline. The comparison against committee-based or extrapolation-grade criteria is argued qualitatively rather than benchmarked empirically, so the claimed cost advantage over ensemble methods is not quantified here. Finally, the tangent-basis construction depends on Gram–Schmidt orthonormalization against a fixed Cartesian reference axis; invariance under in-plane rotations is asserted, but behavior near degenerate spin orientations (where the basis choice becomes ill-conditioned) is not examined.

Conclusion

This work reformulates equivariant evidential uncertainty for magnetic interatomic potentials so that the likelihood, predictive moments, and epistemic covariance share the tangent-plane geometry of the projected spin-force target. The resulting indicator correlates with prediction error at Spearman coefficients near 0.9 and yields simultaneous improvements in energy, force, and projected spin-force accuracy over random sampling on bulk BiFeO$$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$5 and monolayer CrTe$$\mathbf f_s = -\partial E/\partial\hat{\mathbf s}_i$$6. Within its stated scope, the paper establishes tangent-plane-consistent evidential uncertainty as a physically interpretable and data-efficient acquisition signal for magnetic MLIP construction.