Diffusion models achieve state-of-the-art generation quality across many applications, but their ability to capture rare or extreme events in heavy-tailed distributions remains unclear. In this work, we show that traditional diffusion and flow-matching models with standard Gaussian priors fail to capture heavy-tailed behavior. We address this by repurposing the diffusion framework for heavy-tail estimation using multivariate Student-t distributions. We develop a tailored perturbation kernel and derive the denoising posterior based on the conditional Student-t distribution for the backward process. Inspired by $\gamma$-divergence for heavy-tailed distributions, we derive a training objective for heavy-tailed denoisers. The resulting framework introduces controllable tail generation using only a single scalar hyperparameter, making it easily tunable for diverse real-world distributions. As specific instantiations of our framework, we introduce t-EDM and t-Flow, extensions of existing diffusion and flow models that employ a Student-t prior. Remarkably, our approach is readily compatible with standard Gaussian diffusion models and requires only minimal code changes. Empirically, we show that our t-EDM and t-Flow outperform standard diffusion models in heavy-tail estimation on high-resolution weather datasets in which generating rare and extreme events is crucial.

Conditional diffusion and flow models are effective for super-resolving small-scale details in natural images. However, in physical sciences such as weather, three major challenges arise: (i) spatially misaligned input-output distributions (PDEs at different resolutions lead to divergent trajectories), (ii) misaligned and distinct input-output channels (channel synthesis), (iii) several channels with diverse stochasticity scales (multiscale). To address these, we propose to first encode inputs into a latent base distribution that is closer to the target, then apply Flow Matching to generate small-scale physics. The encoder captures deterministic components, while Flow Matching adds stochastic details. To handle uncertainty in the deterministic part, we inject noise via an adaptive noise scaling mechanism, dynamically adjusted by maximum-likelihood estimates of the encoder’s predictions. Experiments on real-world weather data (including super-resolution from 25 km to 2 km scales in Taiwan) and in synthetic Kolmogorov flow datasets show that our proposed Adaptive Flow Matching (AFM) framework outperforms existing methods and produces better-calibrated ensembles.

Modern climate projections lack adequate spatial and temporal resolution due to computational constraints, leading to inaccuracies in representing critical processes like thunderstorms that occur on the sub-resolution scale. Hybrid methods combining physics with machine learning (ML) offer faster, higher fidelity climate simulations by outsourcing compute-hungry, high-resolution simulations to ML emulators. However, these hybrid physics-ML simulations require domain-specific data and workflows that have been inaccessible to many ML experts. This paper is an extended version of our NeurIPS award-winning ClimSim dataset paper. The ClimSim dataset includes 5.7 billion pairs of multivariate input/output vectors spanning ten years at high temporal resolution, capturing the influence of high-resolution, high-fidelity physics on a host climate simulator's macro-scale state. In this extended version, we introduce a significant new contribution in Section 5, which provides a cross-platform, containerized pipeline to integrate ML models into operational climate simulators for hybrid testing. We also implement various baselines of ML models and hybrid simulators to highlight the ML challenges of building stable, skillful emulators. The data (https://huggingface.co/datasets/LEAP/ClimSim_high-res, also in a low-resolution version at https://huggingface.co/datasets/LEAP/ClimSim_low-res and an aquaplanet version at https://huggingface.co/datasets/LEAP/ClimSim_low-res_aqua-planet) and code (https://leap-stc.github.io/ClimSim and https://github.com/leap-stc/climsim-online) are publicly released to support the development of hybrid physics-ML and high-fidelity climate simulations.

Data-driven algorithms, in particular neural networks, can emulate the effect of sub-grid scale processes in coarse-resolution climate models if trained on high-resolution climate simulations. However, they may violate key physical constraints and lack the ability to generalize outside of their training set. Here, we show that physical constraints can be enforced in neural networks, either approximately by adapting the loss function or to within machine precision by adapting the architecture. As these physical constraints are insufficient to guarantee generalizability, we additionally propose to physically rescale the training and validation data to improve the ability of neural networks to generalize to unseen climates. Code-github.com/theucler/CBRAIN-CAM.