An Energy-Based Conservative-Dissipative Latent Neural Evolution Operator for Magnetization Dynamics

Sebastian Schaffer; Lukas Exl

cs.LGSep 3, 2026

An Energy-Based Conservative-Dissipative Latent Neural Evolution Operator for Magnetization Dynamics

Authors: Sebastian Schaffer, Lukas Exl

Organizations: Math. AI/ML, Wolfgang Pauli Institute, Vienna, Austria · Department of Mathematics, University of Vienna, Vienna, Austria

Abstract

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 $μ$ MAG 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

Panel of Figure 1 — 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	$100\times 25\times 1\times 3$	0
Encoder block 1	$49\times 11\times 1\times 16$	1,488
Encoder block 2	$23\times 4\times 1\times 32$	8,768
Encoder block 3	$10\times 1\times 1\times 64$	34,944
Encoder block 4	$4\times 1\times 1\times 128$	57,600
Latent $1\times 1\times 1$ convolution	$4\times 1\times 1\times 128$	16,384

Table 1: Implemented autoencoder. Counts include biases where present.

Model	parameters	RMSE for $\mathcal{T}_{\mathrm{test}}^{k}$	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 (

20.00\text{\,}\mathrm{m}\mathrm{T}29.95\text{\,}\mathrm{m}\mathrm{T}

170^{\circ}

). A–D denotes antisymmetric–dissipative and D dissipative-only. We report the RMSE for the windowed dataset

\mathcal{T}_{\mathrm{test}}^{k}

, and RMSE, MAnE, mean inference time (encoding, full rollout, and decoding) as well as required solver steps (accepted and rejected) for

\mathcal{T}_{\mathrm{test}}

Model	parameters	RMSE for $\mathcal{T}_{\mathrm{test}}^{k}$	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 (

30.00\text{\,}\mathrm{m}\mathrm{T}39.95\text{\,}\mathrm{m}\mathrm{T}

190^{\circ}

Figure 2: Uninterrupted dissipative-only deep–quadratic rollout at

25\text{\,}\mathrm{m}\mathrm{T}

170^{\circ}

, on the

100\times 25\times 1

grid. Solid curves are reference spatial means and dashed curves are decoded-model spatial means.

Figure 3: As Figure 2 , at

36\text{\,}\mathrm{m}\mathrm{T}

190^{\circ}

Figure 4: Post-hoc test-leading antisymmetric–dissipative deep–quadratic rollout at

25\text{\,}\mathrm{m}\mathrm{T}

170^{\circ}

: spatially averaged magnetization, RMSE,

g_{\eta}(t)-g_{\eta}(0)

and scaled Gibbs energy of the reference solution. The rollout is evaluated on the

100\times 25\times 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

36\text{\,}\mathrm{m}\mathrm{T}

190^{\circ}

Figure 6: Reference and decoded in-plane magnetization angle for the A–D deep–quadratic model at $25\text{\,}\mathrm{m}\mathrm{T}$ , $170^{\circ}$ , on the $100\times 25\times 1$ grid. Both angle images use the common interval $[-\pi,\pi]$ . 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 $36\text{\,}\mathrm{m}\mathrm{T}$ , $190^{\circ}$ .

Figure 8: Operator eigenvalues for the field 1 A–D deep–quadratic model. The left and middle panels histogram the imaginary parts of the

\bm{J}

eigenvalues and real parts of the

\bm{R}

eigenvalues; the right panel shows the complex spectrum of

\bm{K}=\bm{J}-\bm{R}

. The structure requires purely imaginary or zero eigenvalues for

\bm{J}

, nonnegative real eigenvalues for

\bm{R}

, and nonpositive real parts for eigenvalues of

\bm{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

2\text{\,}\mathrm{n}\mathrm{s}

rollout at

25\text{\,}\mathrm{m}\mathrm{T}

170^{\circ}

. 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

1\text{\,}\mathrm{n}\mathrm{s}

training horizon. Both models encode exact copies of the same fresh full-order reference state.

Figure 13: As Figure 12 , at

36\text{\,}\mathrm{m}\mathrm{T}

190^{\circ}

Latent Dynamics Modeling

Latent dynamics modeling learns compact hidden-state representations and transition rules to predict, explain, and control evolving systems.

See all 254 papers

Rollout-Decoded Reconstruction for Long-Horizon Prediction in Latent World Models
Aug 25, 2026Rishi Shah, Rishav ShresthaLatent World Models Latent Dynamics Modeling
E3A Healthcare
Latent Evolution Operator (LEO): State-to-State Forecasting of Parametric PDEs
May 14, 2025Khalid Rafiq, Wenjing Liao, Aditya G. NairPDE Surrogate Modeling PDE Operator Learning
Department of Mechanical Engineering, University of Nevada, Reno, Reno, USA · School of Mathematics, Georgia Institute of Technology, Atlanta, USA
Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamics
Aug 22, 2026Haoen Feng, Shenglan Yuan, Shirong LinFlow Map Learning Physics-Informed ML
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

Aug 25, 2026J/K move · Enter open · S save

Rollout-Decoded Reconstruction for Long-Horizon Prediction in Latent World Models

Rishi Shah, Rishav Shrestha

E3A Healthcare

Latent World Models Latent Dynamics Modeling Chaotic Time-Series Prediction

Open paper AskPDF arXiv

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 \pm 0.23

6.97 \pm 0.42

time units at an identical 193,568 parameters, a

1.80\times

improvement confirmed on seeds never used in selection and holding in 10 of 10 preregistered configurations at ratios of 1.71-2.50

\times

. 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.

Abstract

Latent Dynamics Modeling

Explore similar work

Rollout-Decoded Reconstruction for Long-Horizon Prediction in Latent World Models

Latent Evolution Operator (LEO): State-to-State Forecasting of Parametric PDEs

Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamics