Christopher Bülte

Christopher Bülte, Lisa Leimenstoll, Melanie Schienle

In this work, we propose a simulation-based estimation approach using generative neural networks to determine dependencies of precipitation maxima and their underlying uncertainty in time and space. Within the common framework of max-stable processes for extremes under temporal and spatial dependence, our methodology allows estimating the process parameters and their respective uncertainty, but also delivers an explicit nonparametric estimate of the spatial dependence through the pairwise extremal coefficient function. We illustrate the effectiveness and robustness of our approach in a thorough finite sample study where we obtain good performance in complex settings for which closed-form likelihood estimation becomes intractable. We use the technique for studying monthly rainfall maxima in Western Germany for the period 2021-2023, which is of particular interest since it contains an extreme precipitation and consecutive flooding event in July 2021 that had a massive deadly impact. Beyond the considered setting, the presented methodology and its main generative ideas also have great potential for other applications.

Christopher Bülte, Philipp Scholl, Gitta Kutyniok

Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of uncertainties inherent in both model and data has so far rarely been taken into account\textemdash{}a critical limitation in complex, chaotic systems such as weather forecasting. In this paper, we introduce the probabilistic neural operator (PNO), a framework for learning probability distributions over the output function space of neural operators. PNO extends neural operators with generative modeling based on strictly proper scoring rules, integrating uncertainty information directly into the training process. We provide a theoretical justification for the approach and demonstrate improved performance in quantifying uncertainty across different domains and with respect to different baselines. Furthermore, PNO requires minimal adjustment to existing architectures, shows improved performance for most probabilistic prediction tasks, and leads to well-calibrated predictive distributions and adequate uncertainty representations even for long dynamical trajectories. Implementing our approach into large-scale models for physical applications can lead to improvements in corresponding uncertainty quantification and extreme event identification, ultimately leading to a deeper understanding of the prediction of such surrogate models.

Christopher Bülte, Yusuf Sale, Timo Löhr et al.

Uncertainty quantification (UQ) is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we introduce a set of axioms to rigorously assess measures of aleatoric, epistemic, and total uncertainty in supervised regression. By utilizing a predictive exponential family, we can generalize commonly used approaches for uncertainty representation and corresponding uncertainty measures. More specifically, we analyze the widely used entropy- and variance-based measures regarding limitations and challenges. Our findings provide a principled foundation for uncertainty quantification in regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.

Christopher Bülte, Sohir Maskey, Philipp Scholl et al.

Climate change is increasing the occurrence of extreme precipitation events, threatening infrastructure, agriculture, and public safety. Ensemble prediction systems provide probabilistic forecasts but exhibit biases and difficulties in capturing extreme weather. While post-processing techniques aim to enhance forecast accuracy, they rarely focus on precipitation, which exhibits complex spatial dependencies and tail behavior. Our novel framework leverages graph neural networks to post-process ensemble forecasts, specifically modeling the extremes of the underlying distribution. This allows to capture spatial dependencies and improves forecast accuracy for extreme events, thus leading to more reliable forecasts and mitigating risks of extreme precipitation and flooding.

Christopher Bülte, Yusuf Sale, Gitta Kutyniok et al.

Regression tasks, notably in safety-critical domains, require proper uncertainty quantification, yet the literature remains largely classification-focused. In this light, we introduce a family of measures for total, aleatoric, and epistemic uncertainty based on proper scoring rules, with a particular emphasis on kernel scores. The framework unifies several well-known measures and provides a principled recipe for designing new ones whose behavior, such as tail sensitivity, robustness, and out-of-distribution responsiveness, is governed by the choice of kernel. We prove explicit correspondences between kernel-score characteristics and downstream behavior, yielding concrete design guidelines for task-specific measures. Extensive experiments demonstrate that these measures are effective in downstream tasks and reveal clear trade-offs among instantiations, including robustness and out-of-distribution detection performance.

Carlo Kneissl, Christopher Bülte, Philipp Scholl et al.

Probabilistic regression models the entire predictive distribution of a response variable, offering richer insights than classical point estimates and directly allowing for uncertainty quantification. While diffusion-based generative models have shown remarkable success in generating complex, high-dimensional data, their usage in general regression tasks often lacks uncertainty-related evaluation and remains limited to domain-specific applications. We propose a novel diffusion-based framework for probabilistic regression that learns predictive distributions in a nonparametric way. More specifically, we propose to model the full distribution of the diffusion noise, enabling adaptation to diverse tasks and enhanced uncertainty quantification. We investigate different noise parameterizations, analyze their trade-offs, and evaluate our framework across a broad range of regression tasks, covering low- and high-dimensional settings. For several experiments, our approach shows superior performance against existing baselines, while delivering calibrated uncertainty estimates, demonstrating its versatility as a tool for probabilistic prediction.