An energy-based conservative-dissipative latent neural evolution operator for magnetization dynamics

Abstract We introduce a reduced-order learning framework that couples a convolutional autoencoder to a structured latent neural ordinary differential equation. A learned scalar generates the latent dynamics through an antisymmetric operator and a positive-definite dissipative operator. The antisymmetric component permits motion along scalar level sets, while the dissipative component decreases the scalar. Adding a coercive quadratic term gives bounded continuous-time latent trajectories and approach to the scalar's critical set under the stated assumptions. The scalar provides a structural constraint rather than an approximation of physical energy. The encoder, decoder, scalar, and operators are trained jointly on short snapshot windows without time-derivative supervision, physical-energy labels, or dissipation penalties. At inference, one initial state is encoded and evolved continuously in latent space, with decoding only at requested times. We demonstrate the framework on micromagnetic magnetization trajectories from NIST μMAG Standard Problem 4 and compare it with dissipative-only dynamics and an unstructured latent neural differential equation. Repeated training runs show competitive observed-horizon accuracy, while selected longer rollouts illustrate improved retention of approximately stationary states relative to the unstructured baseline. These results motivate scalar-generated latent dynamics as an inductive bias for reduced-order prediction and other learning tasks where bounded evolution and equilibrium behavior are important.