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The paper introduces a novel approach for irregular time series modeling by replacing Neural ODEs with a linear damped harmonic oscillator analogy that admits a closed-form solution, thereby avoiding computationally expensive numerical solvers. Keys and values are modeled as damped, driven oscillators, and the query is expanded in a sinusoidal basis, with attention modeled as a resonance phenomenon. The method is proven to maintain the universal approximation property of continuous-time attention and achieves state-of-the-art performance on irregular time series benchmarks with significant speedups.
Achieve state-of-the-art performance on irregular time series benchmarks while being orders of magnitude faster by ditching Neural ODEs for damped harmonic oscillators with closed-form solutions.
Transformers excel at time series modelling through attention mechanisms that capture long-term temporal patterns. However, they assume uniform time intervals and therefore struggle with irregular time series. Neural Ordinary Differential Equations (NODEs) effectively handle irregular time series by modelling hidden states as continuously evolving trajectories. ContiFormers arxiv:2402.10635 combine NODEs with Transformers, but inherit the computational bottleneck of the former by using heavy numerical solvers. This bottleneck can be removed by using a closed-form solution for the given dynamical system - but this is known to be intractable in general! We obviate this by replacing NODEs with a novel linear damped harmonic oscillator analogy - which has a known closed-form solution. We model keys and values as damped, driven oscillators and expand the query in a sinusoidal basis up to a suitable number of modes. This analogy naturally captures the query-key coupling that is fundamental to any transformer architecture by modelling attention as a resonance phenomenon. Our closed-form solution eliminates the computational overhead of numerical ODE solvers while preserving expressivity. We prove that this oscillator-based parameterisation maintains the universal approximation property of continuous-time attention; specifically, any discrete attention matrix realisable by ContiFormer's continuous keys can be approximated arbitrarily well by our fixed oscillator modes. Our approach delivers both theoretical guarantees and scalability, achieving state-of-the-art performance on irregular time series benchmarks while being orders of magnitude faster.