StocBench: A Benchmark for Generative Modeling of Stochastic Dynamics

Probabilistic forecasting of stochastic dynamical systems, such as turbulent fluid flows, requires models to capture the conditional distribution of future states rather than a single deterministic prediction. While various transport-based generative models—such as diffusion models and flow matching—have been applied to these tasks, their performance under constrained inference budgets remains poorly understood. StocBench addresses this by benchmarking these models on a two-dimensional Kolmogorov flow with stochastic forcing. The study evaluates models on two tasks: a stochastic variant where forcing is unobserved, and a deterministic control task where forcing is provided as input. The benchmark assesses one-step distributional accuracy and the preservation of the invariant measure during long-horizon autoregressive rollouts.

Performance on the stochastic task varies significantly by inference budget. Flow matching (FM) emerges as the most accurate method at high inference budgets, providing the best one-step conditional distribution and maintaining the enstrophy spectrum—a measure of energy distribution across spatial scales—most effectively during rollouts. In the low-budget regime, the second-order exponential integrator DPM-2 is the strongest multi-step method. Distillation-based methods, such as Adversarial Diffusion Distillation (ADD-FM), are highly competitive, matching multi-step performance with only a single network evaluation.

Crucially, the study finds that model performance does not translate between the stochastic and deterministic settings. In the deterministic control task, distillation methods exhibit higher residual variability and larger errors compared to multi-step methods. This suggests that the predictive variability observed in distilled models during stochastic tasks is partly epistemic—stemming from imperfect learning—rather than purely reflecting the irreducible aleatoric uncertainty of the stochastic forcing.

This benchmark provides a standardized, reproducible framework for researchers to evaluate generative models for stochastic PDEs. By isolating the generative step and comparing transport-based models against distillation variants, the study clarifies the trade-offs between computational cost, distributional fidelity, and long-term physical consistency. The results highlight that there is no single dominant method, and practitioners must select models based on their specific inference budget and the nature of the underlying dynamics.