Stochasticity and probabilistic trajectory scoring are essential for data-driven closures of chaotic systems

TL;DR

This paper demonstrates that stochastic modeling and trajectory-based calibration are crucial for accurately capturing the long-term behavior of chaotic systems in data-driven coarse-grained models.

Contribution

It provides a theoretical framework showing the limitations of deterministic losses and proves the effectiveness of stochastic closures calibrated with proper scoring rules.

Findings

Deterministic trajectory training suppresses predictive variance.

Proper scoring rules align forecasts with invariant measures.

Stochastic closures outperform deterministic ones in long-term predictions.

Abstract

Coarse-grained models of chaotic systems neglect unresolved degrees of freedom, inducing structured model error that limits predictability and distorts long-term statistics. Typical data-driven closures are trained to minimize error over a single time step, implicitly assuming Markovian dynamics and often failing to capture long-term behavior. Recent approaches instead optimize losses over finite trajectories. However, when such trajectory-based training is carried out with deterministic pointwise losses, it introduces a fundamental mathematical degeneracy. We prove that optimizing pointwise deterministic losses such as mean squared error over chaotic trajectories suppresses predictive variance, with corresponding loss of physical variability in long integrations. In contrast, strictly proper scoring rules avoid this degeneracy. By targeting forecast distributions rather than realized trajectories, they remove the penalty against predictive spread and align the long-lead optimum with the invariant measure. Using quasi-geostrophic turbulence as a canonical chaotic system, we validate this theory: one-step-trained closures fail to capture stable coarse-grained dynamics, while deterministic closures optimized over trajectories exhibit the variance-loss tendency predicted by our analysis. Stochastic closures calibrated over trajectories using the energy score, however, overcome both structural limitations, yielding skillful ensemble forecasts and realistic long-term statistics. Our results establish that both stochastic modeling and trajectory-based calibration are essential for faithfully representing the dynamics of coarse-grained systems.