Christian Lessig

Antoine Blanchard, Nishant Parashar, Boyko Dodov et al.

Weather extremes are a major societal and economic hazard, claiming thousands of lives and causing billions of dollars in damage every year. Under climate change, their impact and intensity are expected to worsen significantly. Unfortunately, general circulation models (GCMs), which are currently the primary tool for climate projections, cannot characterize weather extremes accurately. To address this, we present a multi-resolution deep-learning framework that, firstly, corrects a GCM's biases by matching low-order and tail statistics of its output with observations at coarse scales; and secondly, increases the level of detail of the debiased GCM output by reconstructing the finer scales as a function of the coarse scales. We use the proposed framework to generate statistically realistic realizations of the climate over Western Europe from a simple GCM corrected using observational atmospheric reanalysis. We also discuss implications for probabilistic risk assessment of natural disasters in a changing climate.

Renu Singh, Robert Brunstein, Antonia Jost et al.

We evaluate the climate simulation capabilities of ArchesWeather and ArchesWeatherGen, two machine learning models originally trained for weather forecasting and evaluated up to a 10-day lead time. ArchesWeather is a deterministic model, while ArchesWeatherGen is a probabilistic flow-matching model leveraging ArchesWeather's forecasts, enabling ensemble-based uncertainty quantification. In this work, we adapt these models to act as forced atmospheric models by using additional conditioning on the monthly mean sea surface temperature (SST) and sea ice cover (SIC) as boundary conditions. In particular, we follow the AI Model Intercomparison Project (AIMIP) Phase 1 protocol, which, analogous to the Atmospheric Model Intercomparison Project (AMIP), proposes a standardized experimental setup to evaluate the climate skill of ML-based forced atmospheric models. We present a comprehensive evaluation of both models under these conditions, including comparison against numerical climate models, ablation studies that examine key design choices in the extension, and an analysis of forced versus unforced configurations. Despite being originally developed for weather forecasting, we demonstrate that forced configurations of ArchesWeather and ArchesWeatherGen produce stable long-term climate simulations, have a stable annual cycle, and capture the drift of many climate variables. The models faithfully reproduce ERA5's climatology, large-scale circulations and interannual variability, and they capture the tails of the distributions.

Thomas Rackow, Nikolay Koldunov, Christian Lessig et al.

Data-driven machine learning models for weather forecasting have made transformational progress in the last 1-2 years, with state-of-the-art ones now outperforming the best physics-based models for a wide range of skill scores. Given the strong links between weather and climate modelling, this raises the question whether machine learning models could also revolutionize climate science, for example by informing mitigation and adaptation to climate change or to generate larger ensembles for more robust uncertainty estimates. Here, we show that current state-of-the-art machine learning models trained for weather forecasting in present-day climate produce skillful forecasts across different climate states corresponding to pre-industrial, present-day, and future 2.9K warmer climates. This indicates that the dynamics shaping the weather on short timescales may not differ fundamentally in a changing climate. It also demonstrates out-of-distribution generalization capabilities of the machine learning models that are a critical prerequisite for climate applications. Nonetheless, two of the models show a global-mean cold bias in the forecasts for the future warmer climate state, i.e. they drift towards the colder present-day climate they have been trained for. A similar result is obtained for the pre-industrial case where two out of three models show a warming. We discuss possible remedies for these biases and analyze their spatial distribution, revealing complex warming and cooling patterns that are partly related to missing ocean-sea ice and land surface information in the training data. Despite these current limitations, our results suggest that data-driven machine learning models will provide powerful tools for climate science and transform established approaches by complementing conventional physics-based models.

Christian Lessig, Ilaria Luise, Bing Gong et al.

The atmosphere affects humans in a multitude of ways, from loss of life due to adverse weather effects to long-term social and economic impacts on societies. Computer simulations of atmospheric dynamics are, therefore, of great importance for the well-being of our and future generations. Here, we propose AtmoRep, a novel, task-independent stochastic computer model of atmospheric dynamics that can provide skillful results for a wide range of applications. AtmoRep uses large-scale representation learning from artificial intelligence to determine a general description of the highly complex, stochastic dynamics of the atmosphere from the best available estimate of the system's historical trajectory as constrained by observations. This is enabled by a novel self-supervised learning objective and a unique ensemble that samples from the stochastic model with a variability informed by the one in the historical record. The task-independent nature of AtmoRep enables skillful results for a diverse set of applications without specifically training for them and we demonstrate this for nowcasting, temporal interpolation, model correction, and counterfactuals. We also show that AtmoRep can be improved with additional data, for example radar observations, and that it can be extended to tasks such as downscaling. Our work establishes that large-scale neural networks can provide skillful, task-independent models of atmospheric dynamics. With this, they provide a novel means to make the large record of atmospheric observations accessible for applications and for scientific inquiry, complementing existing simulations based on first principles.

Anthony McNally, Christian Lessig, Peter Lean et al.

Skilful Machine Learned weather forecasts have challenged our approach to numerical weather prediction, demonstrating competitive performance compared to traditional physics-based approaches. Data-driven systems have been trained to forecast future weather by learning from long historical records of past weather such as the ECMWF ERA5. These datasets have been made freely available to the wider research community, including the commercial sector, which has been a major factor in the rapid rise of ML forecast systems and the levels of accuracy they have achieved. However, historical reanalyses used for training and real-time analyses used for initial conditions are produced by data assimilation, an optimal blending of observations with a physics-based forecast model. As such, many ML forecast systems have an implicit and unquantified dependence on the physics-based models they seek to challenge. Here we propose a new approach, training a neural network to predict future weather purely from historical observations with no dependence on reanalyses. We use raw observations to initialise a model of the atmosphere (in observation space) learned directly from the observations themselves. Forecasts of crucial weather parameters (such as surface temperature and wind) are obtained by predicting weather parameter observations (e.g. SYNOP surface data) at future times and arbitrary locations. We present preliminary results on forecasting observations 12-hours into the future. These already demonstrate successful learning of time evolutions of the physical processes captured in real observations. We argue that this new approach, by staying purely in observation space, avoids many of the challenges of traditional data assimilation, can exploit a wider range of observations and is readily expanded to simultaneous forecasting of the full Earth system (atmosphere, land, ocean and composition).

Christian Lessig

Recent work introduced a unified framework for steerable and directional wavelets in two and three dimensions that ensures many desirable properties, such as a multi-scale structure, fast transforms, and a flexible angular localization. We show that, for an appropriate choice for the radial window function, these wavelets also have closed form expressions for, among other things, the spatial representation, the filter taps for the fast transform, and the frame representation of the Laplace operator. The numerical practicality and benefits of our work are demonstrated using signal estimation from non-uniform, point-wise samples, as required for example in ray tracing, and for reconstructing a signal over a lower-dimensional sub-manifold, with applications for instance in medical imaging.

Paula Harder, Christian Lessig, Matthew Chantry et al.

Deep learning offers promising capabilities for the statistical downscaling of climate and weather forecasts, with generative approaches showing particular success in capturing fine-scale precipitation patterns. However, most existing models are region-specific, and their ability to generalize to unseen geographic areas remains largely unexplored. In this study, we evaluate the generalization performance of generative downscaling models across diverse regions. Using a global framework, we employ ERA5 reanalysis data as predictors and IMERG precipitation estimates at $0.1^\circ$ resolution as targets. A hierarchical location-based data split enables a systematic assessment of model performance across 15 regions around the world.

Christian Lessig

We introduce $Ψ\mathrm{ec}$, a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential $r$-form wavelets with flexible directional localization that provide tight frames for the spaces $Ω^r(\mathbb{R}^n)$ of forms in $\mathbb{R}^2$ and $\mathbb{R}^3$. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the $k$-dimensional integral of an $r$-form is an $(r-k)$-form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of $Ψ\mathrm{ec}$ builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.

Guillaume Couairon, Renu Singh, Anastase Charantonis et al.

Weather forecasting plays a vital role in today's society, from agriculture and logistics to predicting the output of renewable energies, and preparing for extreme weather events. Deep learning weather forecasting models trained with the next state prediction objective on ERA5 have shown great success compared to numerical global circulation models. However, for a wide range of applications, being able to provide representative samples from the distribution of possible future weather states is critical. In this paper, we propose a methodology to leverage deterministic weather models in the design of probabilistic weather models, leading to improved performance and reduced computing costs. We first introduce \textbf{ArchesWeather}, a transformer-based deterministic model that improves upon Pangu-Weather by removing overrestrictive inductive priors. We then design a probabilistic weather model called \textbf{ArchesWeatherGen} based on flow matching, a modern variant of diffusion models, that is trained to project ArchesWeather's predictions to the distribution of ERA5 weather states. ArchesWeatherGen is a true stochastic emulator of ERA5 and surpasses IFS ENS and NeuralGCM on all WeatherBench headline variables (except for NeuralGCM's geopotential). Our work also aims to democratize the use of deterministic and generative machine learning models in weather forecasting research, with academic computing resources. All models are trained at 1.5° resolution, with a training budget of $\sim$9 V100 days for ArchesWeather and $\sim$45 V100 days for ArchesWeatherGen. For inference, ArchesWeatherGen generates 15-day weather trajectories at a rate of 1 minute per ensemble member on a A100 GPU card. To make our work fully reproducible, our code and models are open source, including the complete pipeline for data preparation, training, and evaluation, at https://github.com/INRIA/geoarches .

Guillaume Couairon, Christian Lessig, Anastase Charantonis et al.

One of the guiding principles for designing AI-based weather forecasting systems is to embed physical constraints as inductive priors in the neural network architecture. A popular prior is locality, where the atmospheric data is processed with local neural interactions, like 3D convolutions or 3D local attention windows as in Pangu-Weather. On the other hand, some works have shown great success in weather forecasting without this locality principle, at the cost of a much higher parameter count. In this paper, we show that the 3D local processing in Pangu-Weather is computationally sub-optimal. We design ArchesWeather, a transformer model that combines 2D attention with a column-wise attention-based feature interaction module, and demonstrate that this design improves forecasting skill. ArchesWeather is trained at 1.5° resolution and 24h lead time, with a training budget of a few GPU-days and a lower inference cost than competing methods. An ensemble of four of our models shows better RMSE scores than the IFS HRES and is competitive with the 1.4° 50-members NeuralGCM ensemble for one to three days ahead forecasting. Our code and models are publicly available at https://github.com/gcouairon/ArchesWeather.

Paula Harder, Luca Schmidt, Francis Pelletier et al. · mila

Earth System Models (ESM) are our main tool for projecting the impacts of climate change. However, running these models at sufficient resolution for local-scale risk-assessments is not computationally feasible. Deep learning-based super-resolution models offer a promising solution to downscale ESM outputs to higher resolutions by learning from data. Yet, due to regional variations in climatic processes, these models typically require retraining for each geographical area-demanding high-resolution observational data, which is unevenly available across the globe. This highlights the need to assess how well these models generalize across geographic regions. To address this, we introduce RainShift, a dataset and benchmark for evaluating downscaling under geographic distribution shifts. We evaluate state-of-the-art downscaling approaches including GANs and diffusion models in generalizing across data gaps between the Global North and Global South. Our findings reveal substantial performance drops in out-of-distribution regions, depending on model and geographic area. While expanding the training domain generally improves generalization, it is insufficient to overcome shifts between geographically distinct regions. We show that addressing these shifts through, for example, data alignment can improve spatial generalization. Our work advances the global applicability of downscaling methods and represents a step toward reducing inequities in access to high-resolution climate information.

Sebastian Hoffmann, Christian Lessig

Representation learning has proven to be a powerful methodology in a wide variety of machine learning applications. For atmospheric dynamics, however, it has so far not been considered, arguably due to the lack of large-scale, labeled datasets that could be used for training. In this work, we show that the difficulty is benign and introduce a self-supervised learning task that defines a categorical loss for a wide variety of unlabeled atmospheric datasets. Specifically, we train a neural network on the simple yet intricate task of predicting the temporal distance between atmospheric fields from distinct but nearby times. We demonstrate that training with this task on ERA5 reanalysis leads to internal representations capturing intrinsic aspects of atmospheric dynamics. We do so by introducing a data-driven distance metric for atmospheric states. When employed as a loss function in other machine learning applications, this Atmodist distance leads to improved results compared to the classical $\ell_2$-loss. For example, for downscaling one obtains higher resolution fields that match the true statistics more closely than previous approaches and for the interpolation of missing or occluded data the AtmoDist distance leads to results that contain more realistic fine scale features. Since it is derived from observational data, AtmoDist also provides a novel perspective on atmospheric predictability.

Sebastian Hoffmann, Christian Lessig

The prediction of future climate scenarios under anthropogenic forcing is critical to understand climate change and to assess the impact of potentially counter-acting technologies. Machine learning and hybrid techniques for this prediction rely on informative metrics that are sensitive to pertinent but often subtle influences. For atmospheric dynamics, a critical part of the climate system, no well established metric exists and visual inspection is currently still often used in practice. However, this "eyeball metric" cannot be used for machine learning where an algorithmic description is required. Motivated by the success of intermediate neural network activations as basis for learned metrics, e.g. in computer vision, we present a novel, self-supervised representation learning approach specifically designed for atmospheric dynamics. Our approach, called AtmoDist, trains a neural network on a simple, auxiliary task: predicting the temporal distance between elements of a randomly shuffled sequence of atmospheric fields (e.g. the components of the wind field from reanalysis or simulation). The task forces the network to learn important intrinsic aspects of the data as activations in its layers and from these hence a discriminative metric can be obtained. We demonstrate this by using AtmoDist to define a metric for GAN-based super resolution of vorticity and divergence. Our upscaled data matches both visually and in terms of its statistics a high resolution reference closely and it significantly outperform the state-of-the-art based on mean squared error. Since AtmoDist is unsupervised, only requires a temporal sequence of fields, and uses a simple auxiliary task, it has the potential to be of utility in a wide range of applications.

Christian Lessig

Light field cameras provide intriguing possibilities, such as post-capture refocus or the ability to synthesize images from novel viewpoints. This comes, however, at the price of significant storage requirements. Compression techniques can be used to reduce these but refocusing and reconstruction require so far again a dense pixel representation. To avoid this, we introduce local Fourier slice photography that allows for refocused image reconstruction directly from a sparse wavelet representation of a light field, either to obtain an image or a compressed representation of it. The result is made possible by wavelets that respect the "slicing's" intrinsic structure and enable us to derive exact reconstruction filters for the refocused image in closed form. Image reconstruction then amounts to applying these filters to the light field's wavelet coefficients, and hence no reconstruction of a dense pixel representation is required. We demonstrate that this substantially reduces storage requirements and also computation times. We furthermore analyze the computational complexity of our algorithm and show that it scales linearly with the size of the reconstructed region and the non-negligible wavelet coefficients, i.e. with the visual complexity.

Christian Lessig

We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and an intuitive correspondence to common flow phenomena. Our construction also allows for well defined directional selectivity, e.g. to model the behavior of divergence free vector fields in the vicinity of boundaries or to represent highly directional features like in a von Kármán vortex street. We demonstrate the practicality and efficiency of our construction by analyzing the representation of different divergence free vector fields in our wavelets.

Christian Lessig

We present a local Fourier slice equation that enables local and sparse projection of a signal. Our result exploits that a slice in frequency space is an iso-parameter set in spherical coordinates. Therefore, the projection of suitable wavelets defined separably in these coordinates can be computed analytically, yielding a sequence of wavelets closed under projection. Our local Fourier slice equation then realizes projection as reconstruction with "sliced" wavelets with computational costs that scale linearly in the complexity of the projected signal. We numerically evaluate the performance of our local Fourier slice equation for synthetic test data and tomographic reconstruction, demonstrating that locality and sparsity can significantly reduce computation times and memory requirements.