WaveOperator RNN: Learning Structured World Dynamics with Trainable Local Wave Propagation

Many sequential data are not only temporally ordered but also organized by an underlying topology in which neighboring variables interact locally and information propagates across structured states. Existing sequence models capture temporal or global interactions effectively but rarely encode externally defined local topology explicitly, while neural operators remain largely field-to-field models. We introduce WaveOperator RNN (WO-RNN), which uses local wave propagation as a recurrent operator to combine structured spatial interaction with temporal persistence. Instead of the staggered-grid discretization commonly used for wave equations, WO-RNN adopts a backward-gradient/forward-divergence scheme that preserves local wave coupling in an aligned tensor form. Spatially varying propagation and damping parameters enable heterogeneous, location-dependent dynamics and task-adaptive state evolution. This inductive bias is well suited to sequential and spatiotemporal data with meaningful internal topology. We demonstrate the same operator on tonotopically organized spiking audio and PDE forecasting, illustrating its applicability to sequence modeling and structured world-state evolution. Here, we use world dynamics in the broad sense of predicting how structured latent or physical states evolve through local interactions over time.