Continuous-Time Attention: PDE-Guided Mechanisms for Long-Sequence Transformers

We present Continuous-Time Attention, a novel framework that infuses partial differential equations (PDEs) into the Transformer's attention mechanism to better handle long sequences. Instead of relying on a static attention matrix, we allow attention weights to evolve along a pseudo-time dimension governed by diffusion, wave, or reaction-diffusion dynamics. This dynamic process systematically smooths local noise, strengthens long-range dependencies, and improves gradient stability during training.Our theoretical analysis shows that PDE-driven attention mitigates the exponential decay of distant interactions and improves the optimization landscape. Empirically, Continuous-Time Attention achieves consistent performance gains over both standard and long-sequence Transformer variants across a range of tasks. These results suggest that embedding continuous-time dynamics into attention mechanisms is a promising direction for enhancing global coherence and scalability in Transformer models.