Plausible States Are Not Plausible Trajectories: Self-Supervised Objectives for Chaotic Dynamics

Forecasting provides a natural self-supervised objective for learning chaotic dynamics. Beyond the predictability horizon, however, matching a particular future becomes uninformative: plausible states need not form plausible trajectories. We study this as an objective-design problem for long-horizon generation by factorizing the future into two stochastic tasks: a Jumper samples plausible distant endpoints, and a Completer generates the paths connecting the observed context to those endpoints. Both are trained with the Energy Score and evaluated with state- and path-space diagnostics. On 2D Kuramoto–Sivashinsky and Kolmogorov flow, we show that two seemingly auxiliary design choices affect the learned trajectory distribution. First, the Completer behaves differently when trained on exact future endpoints versus endpoints sampled by the Jumper. Second, sharing one latent draw across a generated trajectory preserves temporal dependence better than independently resampling noise at each time, despite an unchanged or better training score. Thus, for stochastic trajectory learning, the effective objective is determined not only by the loss, but also by what the model is conditioned on during training and how randomness is shared across time.