HAMSTAR: Hamiltonian Structured Inter-Period Refinement for Long-Term Time Series Forecasting

Periodic structure is a fundamental basis for long-term time series forecasting (LTSF), but real-world periodic patterns are not simple repetitions. Variations in level, amplitude, and phase within one period can accumulate across subsequent periods and reshape future periodic structures. We therefore argue that explicitly modeling inter-period relations is a core challenge in LTSF. Existing methods often handle these relations implicitly or rely on unconstrained mixing, where meaningful dependencies can be entangled with noisy interactions. Simply increasing mixing expressiveness does not resolve this issue. Motivated by this, we address LTSF through gradual state refinement driven by interactions among periods. The key challenge is to reinforce meaningful inter-period dependencies while suppressing noise and maintaining stable representations during repeated refinement. To this end, we draw inspiration from Hamiltonian dynamics and use its split-state formulation and symplectic-style updates as an architectural bias for stable iterative refinement. We propose HAMSTAR (HAMiltonian STructured period-Aligned Representation for time series), a Hamiltonian-inspired framework for structured period-aligned time series forecasting. HAMSTAR represents time series as period-aligned latent states, decomposes them into split-state components, and uses structured inter-period coupling to transform period relations into refinement signals. Symplectic-style updates then progressively refine these states, enabling stable representation dynamics. Experiments on standard LTSF benchmarks show that HAMSTAR achieves state-of-the-art performance across diverse datasets, and further analyses demonstrate meaningful inter-period modeling and stable latent dynamics.