We develop an energy-based reduced-order model for micromagnetic magnetization dynamics that couples a convolutional autoencoder to a structured latent neural ordinary differential equation. Motivated by the precessional-dissipative structure of the Landau-Lifshitz-Gilbert equation, the latent vector field is generated from the gradient of a learned scalar potential through an antisymmetric operator and a symmetric positive-semidefinite dissipative operator. This potential is learned in nonunique latent coordinates and is not identified with the Gibbs free energy, but decreases monotonically along autonomous continuous-time solutions, while the antisymmetric component permits motion along its level sets. The encoder, decoder, latent energy, and operators are trained jointly on short trajectory windows using latent and decoded-rollout losses alone, without time-derivative supervision, physical-energy labels, or dissipation penalties. At inference, an initial state is encoded once, evolved in latent space, and decoded only at the requested output times, enabling substantially cheaper trajectory prediction than the micromagnetic solver used to generate the training data. We compare quadratic, deep, and additive deep-quadratic latent energies on two datasets parameterized by field amplitude and generated for the two applied-field directions of the NIST μMAG Standard Problem 4. Dissipative-only and antisymmetric-dissipative models achieve comparable accuracy on short training-style windows but differ substantially on uninterrupted rollouts, for which the antisymmetric-dissipative models provide markedly more accurate trajectory predictions. The deep-quadratic energy gives the best overall accuracy for both field directions and exhibits slower error growth when rollouts are extended to twice the training horizon.
Figures & tables
Figure 1: Overall architecture. The encoder is evaluated at the initial state, the ODE advances the latent space representation, and the decoder supplies requested outputs. Physical parameters condition the latent energy-gradient ODE.
Stage
output tensor
parameters
Input
100×25×1×3
0
Encoder block 1
49×11×1×16
1,488
Encoder block 2
23×4×1×32
8,768
Encoder block 3
10×1×1×64
34,944
Encoder block 4
4×1×1×128
57,600
Latent 1×1×1 convolution
4×1×1×128
16,384
Table 1: Implemented autoencoder. Counts include biases where present.
Model
parameters
RMSE for Ttestk
RMSE
MAnE ( ∘ )
s/trajectory
steps
A–D quadratic
429649
0.0070
0.3201
12.90
0.028
34.0
A–D deep
431314
0.0071
0.2063
6.71
0.047
40.4
A–D deep–quadratic
443666
0.0068
0.0459
1.68
0.073
50.2
D quadratic
413265
0.0091
0.7974
39.10
0.061
99.4
D deep
414930
0.0100
1.4309
64.75
0.040
28.9
D deep–quadratic
427282
0.0084
0.7922
38.26
0.099
70.2
Table 2: Field 1 comparison on 40 held-out trajectories ( 20.00mT29.95mT , 170∘ ). A–D denotes antisymmetric–dissipative and D dissipative-only. We report the RMSE for the windowed dataset Ttestk , and RMSE, MAnE, mean inference time (encoding, full rollout, and decoding) as well as required solver steps (accepted and rejected) for Ttest .
Model
parameters
RMSE for Ttestk
RMSE
MAnE ( ∘ )
s/trajectory
steps
A–D quadratic
429649
0.0411
0.7430
33.17
0.031
37.2
A–D deep
431314
0.0422
0.3363
12.23
0.053
46.1
A–D deep–quadratic
443666
0.0411
0.2929
9.47
0.094
47.4
D quadratic
413265
0.0442
1.3070
77.46
0.042
62.0
D deep
414930
0.0468
0.7632
31.49
0.040
30.0
D deep–quadratic
427282
0.0436
0.6678
29.05
0.095
70.3
Table 3: As Table 2 , for field 2 ( 30.00mT39.95mT , 190∘ ).
Figure 2: Uninterrupted dissipative-only deep–quadratic rollout at 25mT , 170∘ , on the 100×25×1 grid. Solid curves are reference spatial means and dashed curves are decoded-model spatial means.
Figure 3: As Figure 2 , at 36mT , 190∘ .
Figure 4: Post-hoc test-leading antisymmetric–dissipative deep–quadratic rollout at 25mT , 170∘ : spatially averaged magnetization, RMSE, gη(t)−gη(0) and scaled Gibbs energy of the reference solution. The rollout is evaluated on the 100×25×1 grid with relative tolerance 10−3 and absolute tolerance 10−5 . The learned latent energy does not correspond to the Gibbs energy.
Figure 5: As Figure 4 , at 36mT , 190∘ .
Figure 6: Reference and decoded in-plane magnetization angle for the A–D deep–quadratic model at 25mT , 170∘ , on the 100×25×1 grid. Both angle images use the common interval [−π,π] . The right column is the pointwise magnitude of the decoded-minus-reference magnetization vector, with one error scale shared across the four displayed times.
Figure 7: As Figure 6 , at 36mT , 190∘ .
Figure 8: Operator eigenvalues for the field 1 A–D deep–quadratic model. The left and middle panels histogram the imaginary parts of the J eigenvalues and real parts of the R eigenvalues; the right panel shows the complex spectrum of K=J−R . The structure requires purely imaginary or zero eigenvalues for J , nonnegative real eigenvalues for R , and nonpositive real parts for eigenvalues of K .
Figure 9: As Figure 8 , for the field 2 A–D deep–quadratic model.
Figure 10: Training and validation decoded-rollout and latent losses for the field 1 A–D deep–quadratic model.
Figure 11: As Figure 10 , for the field 2 A–D deep–quadratic model.
Figure 12: Paired 2ns rollout at 25mT , 170∘ . The upper panel overlays spatially averaged reference, A–D deep, and A–D deep–quadratic magnetization; color denotes component and line style denotes source. The lower panel shows RMSE for both learned models. The vertical marker denotes the 1ns training horizon. Both models encode exact copies of the same fresh full-order reference state.
A latent world model trains its decoder on latents anchored to observations, then deploys it on the model's own free-running rollout, hundreds of steps past the last observation. Rollout-Decoded Reconstruction (RDR) closes this gap with a single loss term that free-runs the model during training exactly as evaluation will, decodes every rollout latent, and penalizes reconstruction error against ground truth. The term adds no parameters, costs training-time compute only, and reduces to the standard objective at weight zero, so every comparison in this paper is a one-flag A/B. On the chaotic Kuramoto-Sivashinsky equation, RDR raises valid prediction time (the time to first crossing of normalized error 0.5) from 3.87±0.23 to 6.97±0.42 time units at an identical 193,568 parameters, a 1.80× improvement confirmed on seeds never used in selection and holding in 10 of 10 preregistered configurations at ratios of 1.71-2.50×. The results come from a single system; a sweep in which the advantage grows with latent width is descriptive, and control experiments on two classic tasks are preliminary.
Solving time-dependent parametric partial differential equations (PDEs) with high-fidelity solvers is expensive in multi-query settings. This motivates reduced-order models in latent space. Existing methods are typically either (i) recurrent latent dynamics models, which require step-by-step integration and accumulate error over long horizons, or (ii) neural operators, which enable direct prediction but are usually trained from initial conditions and do not explicitly enforce generalization from intermediate states. We introduce the Latent Evolution Operator (LEO), a parameterized map that takes a state at an arbitrary time, together with physical parameters and a forecast horizon, and predicts the future state in a single latent-space evaluation. Under standard manifold and regularity assumptions, finite-time evolution operators admit an encoder-propagator-decoder decomposition. We approximate this structure using neural networks trained on state pairs sampled across anchor times and horizons. We evaluate the method on the 1D Burgers' equation, a 2D advection-diffusion system, and the shallow-water equations. The model produces accurate predictions across parameters and time horizons, achieves significant speedups compared to Autoencoder-LSTM baselines, and maintains accuracy when initialized from intermediate states. These results indicate that learning finite-time evolution operators in latent space provides an effective alternative to recurrent latent dynamics models and operator learning methods trained only from initial conditions.
Khalid Rafiq, Wenjing Liao, Aditya G. Nair
Department of Mechanical Engineering, University of Nevada, Reno, Reno, USA · School of Mathematics, Georgia Institute of Technology, Atlanta, USA
Conventional simulation of current-driven magnetization relies on fine-step integration of the spin-transfer-torque Landau--Lifshitz--Gilbert equation, creating a computational bottleneck in parameter sweeps and control searches. In this work, we propose a physics-constrained neural flow map that learns finite-time dynamics directly on the unit sphere. The model maps the current magnetization, spin-torque strength, and requested time span to a future state in a single forward pass. Tangent-space projection and spherical retraction preserve unit magnetization during recursive, composition-consistent rollout. We validate the framework on single-spin trajectories under in-domain torques and previously unseen but stronger drive. Beyond the training horizon, it achieves an in-domain root mean square error of 0.00425 with norm drift at the 10−7 level. The flow outperforms an adapted Long Short-Term Memory (LSTM) in in-domain accuracy and geometric stability, although the LSTM retains slightly lower out-of-distribution state error. The resulting geometry-preserving propagator reduces reliance on fine-step integration and enables physically admissible long-horizon prediction.
Haoen Feng, Shenglan Yuan, Shirong Lin
School of Physical Sciences, Great Bay University, Dongguan 523000, China · School of Intelligent Systems Science and Engineering, Jinan University, Zhuhai 519000, China · Department of Mathematics, School of Sciences, Great Bay University, Dongguan 523000, China +1