Two dominant approaches have emerged for generating probabilistic forecasts of physical systems: generative models, such as diffusion or flow matching; and ensembles of deterministic models with stochasticity injected, trained using the continuous ranked probability score (CRPS) loss. While both approaches have demonstrated strong predictive accuracy, the reliability of their uncertainties has not been systematically assessed. We address this gap by developing a framework to evaluate both approaches across diverse 2D spatiotemporal physical systems, under matched model size and computational budget. We assess the reliability of probabilistic emulation by inspecting the empirical coverage of predictive intervals, while also considering accuracy and computational efficiency metrics. CRPS-trained ensembles typically achieve more reliable uncertainties on both single-step prediction and autoregressive rollouts, demonstrating better coverage than the standard alternative of training generative models in a latent space. Moreover, the CRPS approach offers significantly faster inference. When generative models are trained in ambient rather than a compressed latent space, which is often infeasible for high-dimensional problems, they exhibit comparable coverage to CRPS-trained ensembles, though with substantially larger inference latency. In contrast, when CRPS-trained ensembles are trained in latent space they do not show a marked degradation in coverage with respect to ambient space. Both generative models and CRPS-trained ensembles demonstrate good predictive accuracy. To facilitate future research and application, we release AutoCast, a modular framework implementing both generative models and CRPS-trained ensembles, alongside AutoSim, a flexible dataset generation package for rapid prototyping.
Many scientific systems exhibit uncertainty from stochastic forcing, unresolved degrees of freedom, or imperfect observations, making reliable surrogate forecasting fundamentally distributional rather than pointwise. For such systems, deterministic neural surrogates fail to capture statistical measures and forecast uncertainty. We introduce TRIE, an evaluation framework for stochastic PDE surrogates that asks whether models reproduce invariant measures, provide trustworthy predictive uncertainty, and scale to efficient probabilistic generation. We demonstrate TRIE on two stationary chaotic spatially extended SPDEs, stochastic Kuramoto--Sivashinsky and stochastic Kolmogorov flow, across 11 parameter values. Our evaluation shows that standard pointwise-trained neural surrogates can produce plausible short rollouts while failing to match long-time statistical structure. Approximate uncertainty methods such as Monte Carlo dropout and heteroscedastic Gaussian likelihoods produce stochastic forecasts, but are often miscalibrated and overconfident under temporal and spatial uncertainty diagnostics. Across these criteria, generative models provide the most consistent performance, accurately capturing invariant measure statistics and achieving the lowest CRPS in all reported probabilistic settings. Finally, we show that latent generative models with automatic dimension discovery retain much of this statistical fidelity while reducing Kolmogorov inference time by roughly 12×. We release our code and data at https://github.com/scailab/TRIE-SPDE-Bench to support reproducible evaluation of stochastic PDE forecasting models.
Bharat Srikishan, Javier E. Santos, Nikhil Muralidhar +1
Machine learning offers a promising avenue to accelerate physical simulations by replacing computationally expensive traditional Partial Differential Equation (PDE) solvers with fast, differentiable surrogate models. However, standard auto-regressive ML emulators often suffer from error accumulation over long horizons and struggle to capture the stochasticity of complex physical systems. In this paper, we propose Kastor, a comprehensive methodology to adapt a deterministic physics foundation model into a highly efficient and accurate generative surrogate. First, we introduce a two-stage inference scheme that combines a large-stride causal auto-regressive model with a non-causal temporal super-resolution network, significantly reducing error accumulation while minimizing computational cost. Second, we present Mean prediction regularization (MPR), a novel training objective that constrains the generative model to predict the deterministic distribution mean under null noise conditioning. This regularization dramatically improves the performance and stability of both Functional Generative Networks (FGN) and diffusion-based emulators. Finally, we demonstrate that incorporating spatial gradient matching improves the accuracy and physical fidelity of the simulations as measured by power spectrum density. Extensive evaluations on diverse simulation datasets of the benchmark The Well show that with these components, our model outperforms competing methods in forecasting accuracy, spectral consistency, and computational efficiency. Our model achieves a 42.9% average reduction in forecasting compared to our reference based on the Walrus finetuning methodology, and outperforms Walrus for 8 out of 10 datasets on variance-normalized RMSE (VRMSE).
Dominant approaches for modelling Partial Differential Equations (PDEs) rely on deterministic predictions, yet many physical systems of interest are inherently chaotic and uncertain. While training probabilistic models from scratch is possible, it is computationally expensive and fails to leverage the significant resources already invested in high-performing deterministic backbones. In this work, we adopt a training-efficient strategy to transform pre-trained deterministic models into probabilistic ones via retrofitting with a proper scoring rule: the Continuous Ranked Probability Score (CRPS). Crucially, this approach is architecture-agnostic: it applies the same adaptation mechanism across distinct model backbones with minimal code modifications. The method proves highly effective across different scales of pre-training: for models trained on single dynamical systems, we achieve 20-54% reductions in rollout CRPS and up to 30% improvements in variance-normalised RMSE (VRMSE) relative to compute-matched deterministic fine-tuning. We further validate our approach on a PDE foundation model, trained on multiple systems and retrofitted on the dataset of interest, to show that our probabilistic adaptation yields an improvement of up to 40% in CRPS and up to 15% in VRMSE compared to deterministic fine-tuning. Validated across diverse architectures and dynamics, our results show that probabilistic PDE modelling need not require retraining from scratch, but can be unlocked from existing deterministic backbones with modest additional training cost.
Cristiana Diaconu, Miles Cranmer, Richard E. Turner +2