Self‑supervised representation learning has advanced time‑series forecasting by extracting robust features from unlabeled data. Existing contrastive or masking approaches often treat time lag as noise or disrupt the temporal dynamics essential for prediction. To overcome these issues, we propose OliO (ODE‑based Linear Transition Operator), a plug‑and‑play self‑supervised framework that learns bidirectional temporal flows as a continuous, structurally consistent dynamical system. OliO introduces a transition operator derived from a linear ordinary differential equation $$\dot{z}(t)=A z(t)$$ where the transition matrix $A$ is constrained to be strictly upper‑triangular. This upper‑triangular constraint mathematically guarantees numerical stability and yields a polynomial boundary that captures long‑term dependencies while preventing the exponential divergence typical of unconstrained ODE models. The resulting structure creates a temporally coherent latent space that preserves the underlying flow needed for accurate forecasting. Extensive experiments across various backbones and benchmarks show that OliO reduces Mean Squared Error (MSE) by up to 5.75% compared with the strongest grid‑searched baselines. Code is released at https://github.com/eai-lab/OliO
Review: OliO explicitly models the continuous flow of time‑series via an upper‑triangular linear ODE, balancing stability and long‑range dependency modeling, marking a significant step forward for self‑supervised forecasting.