Time-Delayed Transformers for Data-Driven Modeling of Low-Dimensional 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.

Exact Sequence Classification with Hardmax Transformers

We prove that hardmax attention transformers perfectly classify datasets of $N$ labeled sequences in $\mathbb{R}^d$, $d\geq 2$. Specifically, given $N$ sequences with an arbitrary but finite length in $\mathbb{R}^d$, we construct a transformer with $\mathcal{O}(N)$ blocks and $\mathcal{O}(Nd)$ parameters perfectly classifying this dataset. Our construction achieves the best complexity estimate to date, independent of the length of the sequences, by innovatively alternating feed-forward and self-attention layers and by capitalizing on the clustering effect inherent to the latter. Our novel constructive method also uses low-rank parameter matrices within the attention mechanism, a common practice in real-life transformer implementations. Consequently, our analysis holds twofold significance: it substantially advances the mathematical theory of transformers and it rigorously justifies their exceptional real-world performance in sequence classification tasks.

Clustering in pure-attention hardmax transformers and its role in sentiment analysis

Transformers are extremely successful machine learning models whose mathematical properties remain poorly understood. Here, we rigorously characterize the behavior of transformers with hardmax self-attention and normalization sublayers as the number of layers tends to infinity. By viewing such transformers as discrete-time dynamical systems describing the evolution of points in a Euclidean space, and thanks to a geometric interpretation of the self-attention mechanism based on hyperplane separation, we show that the transformer inputs asymptotically converge to a clustered equilibrium determined by special points called leaders. We then leverage this theoretical understanding to solve sentiment analysis problems from language processing using a fully interpretable transformer model, which effectively captures `context' by clustering meaningless words around leader words carrying the most meaning. Finally, we outline remaining challenges to bridge the gap between the mathematical analysis of transformers and their real-life implementation.