Coupled Horizons as Operator Coordinates for PDE Trajectory Prediction

Neural PDE surrogates aim to predict finite-time trajectories from observed states. Existing approaches either advance trajectories sequentially or tie joint prediction to a prescribed temporal discretization, making cross-horizon interaction and flexible temporal queries difficult to realize simultaneously. We introduce CURVO, a horizon-coupled neural operator that treats prediction horizons as operator coordinates and represents finite-time evolution through interval-average responses. A multiscale spatial–horizon architecture with shared interval-conditioned refinement couples the queried horizons and reconstructs all requested states from a single observed state in one pass. Trained with a modest fixed query cardinality, the same operator directly evaluates query sets of different cardinalities and placements and supports dense readout of trajectories with up to one hundred states. Across five PDE configurations, CURVO reduces trajectory-averaged error by $4.6$–$37.4\%$ over the strongest one-pass comparator. The formulation further extends to teacher-free one-step image generation with jointly predicted intermediate transport states. CURVO thus makes temporal resolution an inference-time choice while retaining explicit cross-horizon interaction.