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

Martin Thomas Brolly

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.

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

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.

Paper Structure

This paper contains 4 sections, 2 theorems, 64 equations, 8 figures, 1 table.

Table of Contents

  1. The Closure Problem
  2. Data-Driven Closures
  3. Numerical results
  4. Conclusions

Key Result

Proposition 1

Assume further that the coarse-grained state decorrelates under the dynamics in the sense that Denote the MSE loss at lead time $m$ as This objective admits a decomposition of the form where $C_m$ is independent of $\bm{\theta}$. In particular, predictive variance enters the MSE loss as an additive nonnegative term. Under the stated decorrelation assumption, the target conditional mean in the f

Figures (8)

  • Figure 1: Snapshots of upper-layer potential vorticity anomaly in the quasi-geostrophic model at high resolution (left), after coarse graining (center), and from a free-running low-resolution simulation without parameterization (right).
  • Figure 2: Schematic of the generative closures used in the quasi-geostrophic turbulence experiments. This is an example of \ref{['eq:m_tilde']}, where $\bm{G}_{\bm{\theta}}$ is a convolutional neural network, $\overline{\bm{x}}_n$ is the instantaneous potential vorticity anomaly, and $\bm{\xi}$ is a random field with $\operatorname{dim}\bm{\xi}_n=\operatorname{dim}\overline{\bm{x}}_n$ with each component an independent standard Gaussian random variable.
  • Figure 3: Mean energy score as a function of lead time for stochastic closures trained with various window lengths $w$.
  • Figure 4: Mean energy score as a function of lead time for online-trained deterministic and stochastic closures, as well as for the coarse model without a closure.
  • Figure 5: Errors in kinetic energy spectrum as a function of training window length $w$, for deterministic and stochastic closures, as well as without closure.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Proposition 1: Degeneracy of long-lead mean-squared-error training
  • Proposition 2: Asymptotic consistency of strictly proper scoring rules
  • proof : Proof of Proposition 1
  • proof : Proof of Proposition 2
  • Definition 1: Pointwise losses
  • proof : Proof of Supplementary Proposition 1