Towards accurate extreme event likelihoods from diffusion model climate emulators

The gist The authors show that probabilistic information can be accurately recovered for guided samples of Tropical Cyclones using the cBottle diffusion model, allowing them to quantify how much more likely guidance makes a TC sample compared to an unguided model.

How it works

Diffusion models like cBottle emulate present-day climate at high resolution by approximating the probability density of training data through a denoising process (Ho et al., 2020; Song et al., 2020). This process involves iteratively reversing a stochastic noising process where a denoiser network approximates the score function, which is the gradient log-probability density function of the noised data (Dθ(x;σ) − x / σ 2) (1), and sampling is achieved by solving an ODE derived from this process (3).

Extreme event guidance

Guidance in cBottle is implemented by adding a term to the ODE, specifically classifier guidance: dx/dσ = x - Dθ(x,σ)/σ + γ∇xLBCE(C(x),y) (10). This term replaces the score of p(x;σ) with p(x;σy) = p(yx;σ)p(x;σ), where y is a condition such as ‘TC in Miami’ (11). The learning objective of the classifier is −logp(yx;σ), leading to the modified ODE (12): dx/dσ = x - Dθ(x,σ)/σ + σ∇xLBCE(C(x),y) (12).

Importance sampling for likelihoods

To sample extreme events with guidance, the authors use Importance Sampling (IS) by multiplying the samples drawn from the guided distribution by the odds ratio o(x) = p guided(x)/p unguided(x) (15). The log odds ratio is calculated using the PF-ODE trajectory difference between guided and unguided velocity fields: log o(x) = ∆latent + ∆denoiser + ∆guidance (22). This allows them to quantify how much more likely guidance makes us to obtain a TC sample, which can be used to back out the likelihood of the TC under the unguided model (18, 19, 20).

Results and findings

The study finds that setting ρ = 1 and γˆ ∼ 20×64 = 1280 consistently produces guided states without complicated rescaling (4). They find that applying guidance along the entire trajectory leads to implausible values of the guidance divergence term, but restricting guidance to a small window, such as σ ∈ [15,20], is effective (3.1). Furthermore, they demonstrate that for extremes, the uncertainty of the importance sampling estimate can be smaller than that of the unguided Monte Carlo estimate (4).

Applications and limitations

The authors apply these methods to ERA5 samples to calculate likelihoods under different SST input conditions (2.5.1). They note that while guidance and importance sampling are independent of the model itself, cBottle shows a bias where average TC probabilities are too small compared to the historic record (3.4). A significant computational cost is incurred by calculating odds ratios, which can be up to 33x slower than unguided inference for a single guided sample (3.4). Future work suggests exploring classifier-free guidance and improving the model's representation of TC strengths (4).

Conclusion

The authors conclude that they can retrieve probabilistic information from a diffusion model guided to oversample extreme events, and the accurate odds ratios increase their confidence in extreme event guidance being consistent with the data distribution (4). The method is successful for weaker guidance and sampling repeatedly, and it shows potential for more cost-effective sampling of extremes in the long term (3.4). The authors also demonstrate that this probability calculation could be used to evaluate the KL-divergence between two data distributions p and q (3.4).

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Towards accurate extreme event likelihoods from diffusion model climate emulators climate emulators Peter Manshausen, Noah Brenowitz, Julius Berner, Karthik Kashinath, and Mike Pritchard 1NVIDIA, Santa Clara, CA, USA Correspondence: Peter Manshausen (pmanshausen@nvidia.com) Abstract. ML climate model emulators are useful for scenario planning and adaptation allowing for cost-efficient experimentation. Recently the diffusion model Climate in a Bottle (cBottle) has been proposed for generation of atmospheric states compatible with boundary conditions of solar position and sea surface temperatures. Crucially cBottle can be guided to generate extreme events such as Tropical Cyclones (TCs) over locations of interest. Diffusion models such as cBottle work by approximating the probability density of the training data. Here we show use cases of the probability density estimates of atmospheric states obtained from this climate emulator. Most importantly these estimates allow us to calculate likelihoods of extreme events under guidance. When guiding the model towards states including TCs comparing the probability density under the guided and unguided model enables us to quantify how much more likely the guidance has made the TC. We show how these odds ratios allow us to importance-sample from the TC distribution, reducing the standard error of the probability estimate compared to simple Monte Carlo sampling. Furthermore we discuss results and limitations of the application of model probability densities to extreme event attribution-like experiments. We present these early but encouraging results hoping they will spur more research into probabilistic information that can be gained from diffusion models of the atmosphere Copyright statement. CC-BY 1 Introduction Generative Artificial Intelligence has achieved remarkable success across the modalities of natural language, images, and various scientific applications in recent years. One such success is in medium-range weather forecasting (Keisler, 2022) matching and then surpassing (Pathak et al., 2022; Price et al., 2023) the forecast skill of numerical weather prediction and being increasingly deployed operationally (Lang et al., 2014). For climate research has focused on stable rollouts of dynamical (auto-regressive) models which predict a following timestep given a current one. Stability is needed on a much longer timescale than for weather in order to allow for studies into climate variability. Some models have shown promising skill at emulating the present-day climate (Watt-Meyer et al., 2025; Chapman et al., 2025). Another approach to simulating climate is the direct generation of samples without a previous time step as input. This way Watson-Parris et al. (2022) have been able to emulate the behavior of a climate model conditioned on either anthropogenic emissions. Bouabid et al. (2025) use diffusion models to generate impact-relevant variables sampled from the distributions of three CMIP6 models conditioned on global mean surface temperature. Perkins et al. (2026) demonstrate downscaling of 100km climate runs to 3km resolution precipitation fields. Climate-in-a-Bottle (cBottle, Brenowitz et al., 2025) is a recent family of diffusion models that emulate the present-day climate at unprecedented 5km resolution. Conditioned on sea surface temperatures and solar position, the first step coarse model generates full atmospheric states of winds temperature humidity and geopotential at eight pressure levels. Being a diffusion model it has the unique capability for climate emulators to be guided to states of interest in a way similar to the implementation of score based data assimilation (Rozet and Louppe, 2023) with a diffusion weather emulator by Manshausen et al. (2025). Brenowitz et al. (2025) demonstrate this by training a classifier of Tropical Cyclone (TC) probability and using it for classifier guidance allowing a user to generate samples that have a TC in a user-defined location. Such oversampling can be useful to study the impacts of these extremes for scenario planning and building intuition for possible events. However the model does not inform the user about the likelihood of such generated samples. For diffusion models which learn to estimate the probability density function of the training data there are theoretical approaches to calculating the probabilities of samples. Song et al. (2020) detail how these probability estimates can be obtained from the probability flow (PF) ODE associated with the diffusion model. However these PDF values can be counterintuitive with Nalisnick et al. (2019) proving that models can assign higher values to samples from data sets other than the training set. Here we show that nevertheless probabilistic information can be accurately recovered for guided samples of TCs using the cBottle diffusion model. Specifically we can quantify how much more likely guidance makes us to obtain a TC sample which can be used to back out the likelihood of the TC under the unguided model. We further demonstrate other applications of the sample probability calculated from cBottle. The remainder of the paper is organized as follows: Section 2 details the methods used here with subsection 2.1 discussing more general concepts before subsections 2.4 describes the methods related to obtaining TC likelihoods and subsection 2.5 and detail the application to ERA5 samples Section 3 shows our results for guidance TC likelihoods and other applications before the final section concludes

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Song et al., Score-based generative modeling through stochastic differential equations arXiv preprint arXiv:2011.13456, 2020 Song et al., Score-based generative modeling through stochastic differential equations arXiv preprint arXiv:2011.13456, 2020

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The ODE formulation allows us to back out the model’s estimated probability density for individual samples by (Song et al.

1. Accurate Extreme Event Likelihood Estimation: The system can now calculate likelihoods of extreme events, such as Tropical Cyclones (TCs), by quantifying how much more likely the guidance has made the TC by comparing probability density under guided and unguided models, using odds ratios.

2. Importance Sampling for Rare Events: The AI system can perform importance sampling from the TC distribution using these odds ratios, which is shown to reduce the standard error of probability estimates compared to simple Monte Carlo sampling, leading to more confident estimates of extreme event likelihoods for rare events.

3. Adaptive Guidance Strategy: The model can utilize an adaptive guidance scale, specifically γ = ˆγρ, where ρ is defined as ∥ x−Dθ(x,σ)∥2∥∇xLBCE(C(x),y)∥2 to ensure that at each step, the fraction of the norm of the update in the score direction and of that in the classifier direction is constant.

4. Efficient Guidance Window Identification: The system can determine an optimal guidance window for extreme events, finding that it is sufficient to apply guidance at a single sampling step corresponding to the magnitude of the extreme event, specifically within an interval like σ ∈ [15,20].

5. Reduced Computational Cost via Distillation: To mitigate the high computational cost of calculating odds ratios (a slowdown of 33x for a single guided sample), future systems can benefit from distillation of the probability calculation, potentially reducing the overall slowdown to just 2x (one calculation for the guided and one for the unguided model).

6. Model Bias Detection: The system can compare its estimated TC probabilities with historic ERA5 data, identifying that while it matches well spatially, the average values of TC probability are too small in cBottle, indicating a specific bias that could be addressed by improving the representation of extremes in the diffusion model.

ML climate model emulators are useful for scenario planning and adaptation, allowing for cost-efficient experimentation. Recently, the diffusion model Climate in a Bottle (cBottle) has been proposed for generation of atmospheric states compatible with boundary conditions of solar position and sea surface temperatures. Crucially, cBottle can be guided to generate extreme events such as Tropical Cyclones (TCs) over locations of interest. Diffusion models such as cBottle work by approximating the probability density of the training data. Here, we show use cases of the probability density estimates of atmospheric states obtained from this climate emulator. Most importantly, these estimates allow us to calculate likelihoods of extreme events under guidance. When guiding the model towards states including TCs, comparing the probability density under the guided and unguided model enables us to quantify how much more likely the guidance has made the TC. We show how these odds ratios allow us to importance-sample from the TC distribution, reducing the standard error of the probability estimate compared to simple Monte Carlo sampling. Furthermore, we discuss results and limitations of the application of model probability densities to extreme event attribution-like experiments. We present these early but encouraging results hoping they will spur more research into probabilistic information that can be gained from diffusion models of the atmosphere.