SDHilb: Schrödinger Dynamics-Guided Neural Network with Adaptive Multi-Scale Hilbert Transform for Time Series Forecasting

Conventional real-valued neural networks struggle to explicitly capture phase-frequency coupling, while long-horizon time series forecasting requires models to characterize non-stationary amplitude variations, phase shifts, and frequency-dependent temporal evolution. This mismatch limits the stability and expressiveness of neural models on complex forecasting tasks. To address this challenge, we propose SDHilb, a structure-preserving forecasting framework that lifts real-valued observations into an adaptive complex state space and models temporal evolution as a process jointly governed by amplitude and phase information. Through the Hilbert transform, phase-related structures are explicitly encoded, leading to a coupled evolution of real and quadrature components that motivates Schrödinger-Hamiltonian structured dynamics in the lifted complex space. Based on this formulation, SDHilb integrates Schrödinger-guided linear dynamics with an adaptive multi-scale Hilbert transform and introduces three key components: (1) a structure-preserving linear dynamic module with symplectic discretization, (2) an adaptive multi-scale Hilbert module for constructing data-dependent quadrature components and refined time-frequency decomposition, and (3) a theoretically grounded parameter reuse mechanism derived from the closed-form recurrence structure. Extensive experiments demonstrate that SDHilb achieves competitive or superior performance on both short- and long-term forecasting tasks while reducing horizon-specific parameterization through recurrence-guided parameter reuse.