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

Forecasting provides a natural self-supervised objective for learning models of dynamical systems. For chaotic systems, however, individual trajectories become unpredictable beyond a finite predictability horizon, and plausible states need not form plausible trajectories. Beyond this horizon, the objective should shift from predicting a particular future to generating trajectories that preserve both the statistics of physical states and their temporal evolution. We study this as an objective-design problem using a Jump-and-Complete framework that factorizes trajectory generation 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 first show that, past the predictability horizon, pointwise errors fail to distinguish models with very different temporal behavior. We then show that two seemingly auxiliary design choices shape the learned trajectory distribution. First, training the Completer on Jumper-sampled rather than exact endpoints changes which temporal statistics of composed rollouts are matched on Kolmogorov flow. Second, sharing one latent draw across a trajectory preserves temporal dependence better than resampling noise at each time, despite an equal or better Energy 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.