Data-Driven Thiele Equation Approach for State-Dependent Coefficients in the Nonlinear Dynamics of Vortex-based Nano-Oscillators

The gist The data-driven Thiele equation approach extracts effective, position-dependent Thiele coefficients from a single current-ramp micromagnetic simulation by interpolating the magnetization in vortex-core-position space, revealing quantitative signatures of confinement and motion-induced vortex deformation.

Introduction

Magnetic vortices are nonuniform magnetization states in which the in-plane magnetization curls around a nanoscale core The usual coordinate for a magnetic vortex is the vortexcore position Xc, and its corresponding Thiele equation expresses a balance among gyrotropic, dissipative, conservative, and current-induced forces In many practical implementations, these coefficients are assumed to be constant or are assigned simplified analytical forms derived from a rigid-vortex approximation This approximation becomes restrictive in the nonlinear regime Full micromagnetic simulations (MMS) contain these effects without imposing a rigid ansatz, but they do not provide a compact coefficient-level description of the resulting dynamics Machine learning-based models can be predictive while sacrificing the interpretation supplied by a collectivecoordinate equation The central question addressed here is therefore how effective collectivecoordinate descriptions can be systematically extracted from full micromagnetic dynamics while retaining physical interpretability.

State-dependent collective-coordinate framework

The reduced description is constructed by sampling the magnetization over a broad range of vortex-core positions <ref:2508.14829#pg3> For each saved snapshot, the vortex core is located and its position is written as Xc = Xcxˆ + Ycyˆ = sR eˆr(φ) <ref:2508.14829#pg4> The magnetization field m = m(r; Xc) which is regarded as a function of both the spatial coordinate r and the collective coordinate Xc <ref:2508.14829#pg4>. The effective Thiele coefficients are obtained by evaluating the standard Thiele integrands locally on each cell and summing their contributions over the magnetic volume <ref:2508.14829#pg5> Each coefficient Cα can be computed as Cα(Xc) = Z V fα m, ∂m/∂Xc, ∂m/∂Yc dV <ref:2508.14829#pg5>.

State-dependent gyrotropic and dissipative dynamics

The extracted maps reveal a radius-dependent gyrotropic response and an increasing separation between radial and azimuthal dissipation, directly exposing the breakdown of a rigid, constant-coefficient picture <ref:2508.14829#pg6> The gyrotropic coefficient is the effective gyrotropic coupling of the selected core coordinate to the complete, deformable texture <ref:2508.14829#pg7>. The damping tensor exhibits the strongest position dependence, as both components coincide at small orbit radius but progressively separate as the orbit expands <ref:2508.14829#pg7> This separation measures an anisotropy of the collective-coordinate manifold, where a unit radial displacement and a unit azimuthal displacement no longer induce equivalent rearrangements of the magnetization <ref:2508.14829#pg7>.

Dynamics beyond the sampled trajectory

The extracted coefficient functions are intended not only to reproduce the current-ramp data but also to test whether the sampled state manifold defines a transferable collective-coordinate equation <ref:2508.14829#pg8> The conventional constant-coefficient model predicts vortex expulsion, i.e., s ≥ 0.8, at 2.25 µs <ref:2508.14829#pg8>. In contrast, DD-TEA reproduces the overall transient response throughout the entire protocol with a mean-normalized RMSE of 9% for s(t) and 1% for f(t) <ref:2508.14829#pg8>. The remaining discrepancies identify velocity-dependent deformation as a natural target for higher-dimensional coefficient maps or additional collective coordinates <ref:2508.14829#pg8>.

Conclusion

The extracted equation is an effective description tied to the selected collective coordinates, retaining the gyrotropic, dissipative, conservative, and spin-transfer structure of the collective-coordinate equation without imposing a rigid analytical texture <ref:2508.14829#pg8>. The extracted coefficients show that the nonlinear dynamics cannot be reduced to a position-dependent restoring force alone <ref:2508.14829#pg8>. The gyrovector magnitude increases with orbit radius, while the radial and azimuthal damping coefficients evolve in opposite directions, revealing an emergent anisotropy of the collective dissipation <ref:2508.14829#pg8>. Including this state dependence reproduces stable nonlinear orbits and predicts the response to an independent time-varying current protocol, whereas a conventional constant-coefficient description predicts vortex expulsion <ref:2508.14829#pg8>. The resulting reduction in computational cost is substantial, but it follows from the more fundamental result that full micromagnetic dynamics can be converted into an interpretable, state-dependent collective-coordinate equation for a deformable magnetic texture <ref:2508.14829#pg8>.

Supplemental Materials

The training data was generated from a single current-ramp simulation designed to sample a broad range of stable vortex-core positions at low computational cost <ref:2508.14829#pg11> The resulting dataset therefore provides an approximately uniform coverage of the explored (Xc, Yc) domain <ref:2508.14829#pg4>. The optimal configuration is found to be a radial basis function interpolator with a multiquadric kernel, smoothing parameter λ = 10−5, and scale parameter ϵ = 10 <ref:2508.14829#pg4>. The extracted data are accurately described by the polynomial interpolation, yielding a coefficient of determination of R2 > 0.999 <ref:2508.14829#pg4>. Overall, the extracted coefficients reproduce the expected behavior of vortex dynamics in a circular nanodot <ref:2508.14829#pg8>. The stiffness coefficient k0(s) and the Oersted contribution κOe(s) closely follow the trends reported in Ref. [31], while the spin-transfer coefficient remains essentially constant, consistent with the rigid-vortex approximation <ref:2508.14829#pg8>.

Convergence study

The results show a monotonic convergence of the steady-state radius, frequency, and total energy as the mesh is refined <ref:2508.14829#pg8> Based on these results, an in-plane discretization of 2 × 2 nm squared (nx = ny = 100) was selected for all simulations reported in the main text and in the present Supplemental Materials <ref:2508.14829#pg8>. The residual differences relative to finer discretizations are small compared to the variations of the effective coefficients over the explored range of vortex-core positions <ref:2508.14829#pg8>.

References

[1] T. Shinjo et al. “Magnetic Vortex Core Observation in Circular Dots of Permalloy”. In: Science 289.5481 (2000), pp. 930–932 <ref:2508.14829#pg8>

[11] A. A. Thiele. “Steady-State Motion of Magnetic Domains”. In: Phys. Rev. Lett. 30.6 (1973), pp. 230–233 <ref:2508.14829#pg8>

[15] K. Y. Guslienko et al., “Eigenfrequencies of vortex state excitations in magnetic submicronsize disks”. In: Journal of Applied Physics 91.10 (2002), pp. 8037–8039 <ref:2508.14829#pg8>

[25] Je-Ho Shim et al., “Nonlinear motion of magnetic vortex under alternating-current magnetic field: Dynamic correction of a gyrovector and a damping tensor of the Thiele’s equation”. In: Applied Physics Letters 99.14 (Oct. 2011), p. 142505 <ref:2508.14829#pg8>

[36] Konstantin Y. Guslienko, Oksana V. Sukhostavets, and Dmitry V. Berkov, “Nonlinear magnetic vortex dynamics in a circular nanodot excited by spin-polarized current”. In: Nanoscale Research Letters 9 (2014), p. 386 <ref:2508.14829#pg8>

[31] Simon de Wergifosse, Chloe Chopin, and Flavio Abreu Araujo, “Quantitative and realistic description of the magnetic potential energy of spin-torque vortex oscillators”. In: Physical Review B 108.

1. textbf Enhanced Nonlinear Dynamics Prediction for STVOs in Confined Geometries: The DD-TEA System Enables Accurate Trajectory Forecasting. This improved AI system can accurately predict vortex expulsion or reproduce the nonlinear stable-orbit dynamics by incorporating state-dependent coefficients, whereas a conventional constant-coefficient description predict[s] vortex expulsion.

2. textbf Systematic Extraction of Effective Coefficients from Full Micromagnetic States: The DD-TEA Framework Provides Interpretable Dynamics. The system can extract effective, positiondependent Thiele coefficients from a single current-ramp micromagnetic simulation by interpolating the magnetization in vortex-core-position space, allowing for a physically interpretable description rather than acting as a black-box surrogate.

3. textbf Detection of Emergent Anisotropy in Collective Dissipation: The Model Identifies State-Dependent Coupling. The AI can reveal physical insights such as the increasing separation between radial and azimuthal dissipation, which quantifies the departure of the collective dynamics from a rigid, constant-coefficient vortex model by showing how a unit radial displacement and a unit azimuthal displacement no longer induce equivalent rearrangements of the magnetization.

4. textbf Robust Validation Against Independent Simulations: The DD-TEA Ensures Transferability Across Excitation Protocols. The system can be validated against novel protocols where all Thiele coefficients remain fixed to the functions obtained from the single current-ramp simulation, with no additional fitting, demonstrating that the agreement is retained despite reversing the vortex polarity and current sign relative to the training simulation.

5. textbf Identification of Missing Dynamical Degrees of Freedom: The System Diagnoses Model Limitations. By comparing results with full micromagnetic simulations, the system can identify what is omitted, such as velocity-dependent deformation as a natural target for higher-dimensional coefficient maps or additional collective coordinates, guiding the creation of more general predictive models.