Time-delayed transformers for data-driven modeling of low-dimensional dynamics

We propose the time-delayed transformer (TD-TF), a simplified TF architecture for data-driven modeling of unsteady spatio-temporal dynamics. TD-TF bridges linear operator-based methods and deep sequence models by showing that a single-layer, single-head TF can be interpreted as a nonlinear generalization of TD dynamic mode decomposition. The architecture is deliberately minimal, consisting of one self-attention layer with a single query per prediction and one feedforward layer, resulting in linear computational complexity in sequence length and a small parameter count. Numerical experiments demonstrate that TD-TF matches the performance of linear baselines on near-linear systems, while significantly outperforming them in nonlinear and chaotic regimes, where it accurately captures long-term dynamics. Validation studies on synthetic signals, unsteady aerodynamics, the Lorenz’63 system, and a reaction-diffusion model show that TD-TF preserves the interpretability and efficiency of linear models while providing substantially enhanced expressive power for complex dynamics.