Time-Delayed Transformers for Data-Driven Modeling of Low-Dimensional Dynamics
This work addresses modeling complex dynamics in fields like aerodynamics and chaos theory, offering an incremental improvement by bridging linear operator-based methods and deep sequence models.
The paper tackled data-driven modeling of unsteady spatio-temporal dynamics by proposing the time-delayed transformer (TD-TF), a simplified transformer architecture that matches linear baselines on near-linear systems and significantly outperforms them in nonlinear and chaotic regimes, accurately capturing long-term dynamics.
We propose the time-delayed transformer (TD-TF), a simplified transformer 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 transformer can be interpreted as a nonlinear generalization of time-delayed dynamic mode decomposition (TD-DMD). 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 strong 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.