Local climate information is crucial for impact assessment and decision-making, yet coarse global climate simulations cannot capture small-scale phenomena. However, to preserve physical properties, estimating spatio-temporally coherent high-resolution weather dynamics for multiple variables across long time horizons is crucial. We present a novel generative approach that uses a score-based diffusion model trained on high-resolution reanalysis data to capture the statistical properties of local weather dynamics. After training, we condition on coarse climate model data to generate weather patterns consistent with the aggregate information. As this inference task is inherently uncertain, we leverage the probabilistic nature of diffusion models and sample multiple trajectories. We evaluate our approach with high-resolution reanalysis information before applying it to the climate model downscaling task. We then demonstrate that the model generates spatially and temporally coherent weather dynamics that align with global climate output. Numerical simulations based on the Navier-Stokes equations, discretized over time and space, are fundamental to understanding weather patterns, climate variability, and climate change. Stateof-the-art numerical weather prediction (NWP) models, which primarily focus on atmospheric processes, can accurately resolve small-scale dynamics within the Earth system, providing fine-scale spatial and temporal weather patterns at resolutions on the order of kilometers [1]. However, the substantial computational resources required for these models render them impractical for simulating the extended time scales associated with climatic changes. In contrast, Earth System Models (ESMs), such as those included in the CMIP6 project [2], incorporate a broader range of processes--including atmospheric, oceanic, and biogeochemical interactions--while operating on coarser spatial scales. This coarse resolution limits the ability of ESMs to fully capture small-scale processes, requiring parameterizations to represent unresolved dynamics as functions of resolved variables. This work introduces a probabilistic downscaling pipeline that jointly estimates spatio-temporally consistent weather dynamics from ESM simulations on multiple variables. The framework is built around a score-based diffusion model and can be understood as a combination of four modules, which can each be adjusted independently of the others. This schematic outlines the framework.

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving {high-dimensional} ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems -- most importantly, the solution of discretised {partial} differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.

This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.