Trajectory-based probabilistic learning is essential for representing unresolved dynamics in chaotic systems

Coarse-grained models of chaotic systems neglect unresolved degrees of freedom, inducing structured model error that limits predictability and distorts long-term statistics. Standard data-driven closures address this by training offline to minimize one-step prediction error, implicitly assuming Markovian dynamics and deterministic corrections. Here we demonstrate that this paradigm is fundamentally flawed. Using mesoscale turbulence as a canonical multiscale system, we show that offline training yields poorly calibrated forecasts and incorrect stationary statistics, regardless of model complexity. In contrast, stochastic closures trained on trajectories using proper scoring rules recover reliable ensemble forecasts and realistic long-term behavior. We find that this improvement stems not from architectural sophistication, but from probabilistic calibration over multiple time steps. Our results identify online (trajectory-based) learning and stochasticity as structural requirements for representing unresolved dynamics, with significant implications for Earth system modelling and data-driven prediction more broadly.