Publications

2025

2025

1. TMLR

   

   Probabilistic neural operators for functional uncertainty quantification

   Christopher Bülte, Philipp Scholl, and Gitta Kutyniok

   Transactions on Machine Learning Research, 2025

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   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—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.

   @article{bultepno,
     title = {Probabilistic neural operators for functional uncertainty quantification},
     author = {B{\"u}lte, Christopher and Scholl, Philipp and Kutyniok, Gitta},
     journal = {Transactions on Machine Learning Research},
     issn = {2835-8856},
     year = {2025},
     url = {https://openreview.net/forum?id=gangoPXSRw},
     note = {},
   }

2. AIES

   

   Uncertainty quantification for data-driven weather models

   Christopher Bülte, Nina Horat, Julian Quinting, and Sebastian Lerch

   Artificial Intelligence for the Earth Systems, 2025

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   Artificial intelligence (AI)-based data-driven weather forecasting models have experienced rapid progress over the last years. Recent studies, with models trained on reanalysis data, achieve impressive results and demonstrate substantial improvements over state-of-the-art physics-based numerical weather prediction models across a range of variables and evaluation metrics. Beyond improved predictions, the main advantages of data-driven weather models are their substantially lower computational costs and the faster generation of forecasts, once a model has been trained. However, most efforts in data-driven weather forecasting have been limited to deterministic, point-valued predictions, making it impossible to quantify forecast uncertainties, which is crucial in research and for optimal decision making in applications. Our overarching aim is to systematically study and compare uncertainty quantification methods to generate probabilistic weather forecasts from a state-of-the-art deterministic data-driven weather model, Pangu-Weather. Specifically, we compare approaches for quantifying forecast uncertainty based on generating ensemble forecasts via perturbations to the initial conditions, with the use of statistical and machine learning methods for post-hoc uncertainty quantification. In a case study on medium-range forecasts of selected weather variables over Europe, the probabilistic forecasts obtained by using the Pangu-Weather model in concert with uncertainty quantification methods show promising results and provide improvements over ensemble forecasts from the physics-based ensemble weather model of the European Centre for Medium-Range Weather Forecasts for lead times of up to 5 days.

   @article{bulteUncertaintyQuantificationDatadriven2024,
     title = {Uncertainty quantification for data-driven weather models},
     author = {Bülte, Christopher and Horat, Nina and Quinting, Julian and Lerch, Sebastian},
     journal = {Artificial Intelligence for the Earth Systems},
     year = {2025},
     doi = {10.1175/AIES-D-24-0049.1},
     url = {https://journals.ametsoc.org/view/journals/aies/aop/AIES-D-24-0049.1/AIES-D-24-0049.1.xml},
   }

2024

2024

1. NeurIPS 2024

   

   Probabilistic predictions with Fourier neural operators

   Christopher Bülte, Scholl Philipp, and Kutyniok Gitta

   NeurIPS 2024 workshop on Bayesian Decision-making and Uncertainty, 2024

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   Neural networks have been successfully applied in modeling partial differential equations, especially in dynamical systems. Commonly used models, such as neural operators, are performing well at deterministic prediction tasks, but lack a quantification of the uncertainty inherent in many complex systems, for example weather forecasting. In this paper, we explore a new approach that combines Fourier neural operators with generative modeling based on strictly proper scoring rules in order to create well-calibrated probabilistic predictions of dynamical systems. We demonstrate improved predictive uncertainty for our approach, especially in settings with very high inherent uncertainty.

   @article{prob_predictions_neural_operator,
     title = {Probabilistic predictions with Fourier neural operators},
     author = {Bülte, Christopher and Philipp, Scholl and Gitta, Kutyniok},
     journal = {NeurIPS 2024 workshop on Bayesian Decision-making and Uncertainty},
     url = {https://openreview.net/forum?id=orKA6gJwlB},
     year = {2024},
   }

2. arXiv

   

   Estimation of Spatio-Temporal Extremes via Generative Neural Networks

   Christopher Bülte, Lisa Leimenstoll, and Melanie Schienle

   arXiv preprint arXiv:2407.08668, 2024

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   Recent methods in modeling spatial extreme events have focused on utilizing parametric max-stable processes and their underlying dependence structure. In this work, we provide a unified approach for analyzing spatial extremes with little available data by estimating the distribution of model parameters or the spatial dependence directly. By employing recent developments in generative neural networks we predict a full sample-based distribution, allowing for direct assessment of uncertainty regarding model parameters or other parameter dependent functionals. We validate our method by fitting several simulated max-stable processes, showing a high accuracy of the approach, regarding parameter estimation, as well as uncertainty quantification. Additional robustness checks highlight the generalization and extrapolation capabilities of the model, while an application to precipitation extremes across Western Germany demonstrates the usability of our approach in real-world scenarios.

   @article{bulteEstimationSpatiotemporalExtremes2024,
     title = {Estimation of Spatio-Temporal Extremes via Generative Neural Networks},
     author = {Bülte, Christopher and Leimenstoll, Lisa and Schienle, Melanie},
     journal = {arXiv preprint arXiv:2407.08668},
     year = {2024},
     eprint = {2407.08668},
     eprinttype = {arXiv},
     eprintclass = {cs, stat},
     url = {http://arxiv.org/abs/2407.08668},
     urldate = {2024-07-13},
     keywords = {Computer Science - Machine Learning,Statistics - Machine Learning},
   }

2023

2023

1. Energy & AI

   

   Multivariate Time Series Imputation for Energy Data Using Neural Networks

   Christopher Bülte, Max Kleinebrahm, Hasan Ümitcan Yilmaz, and Juan Gómez-Romero

   Energy and AI, 2023

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   Multivariate time series with missing values are common in a wide range of applications, including energy data. Existing imputation methods often fail to focus on the temporal dynamics and the cross-dimensional correlation simultaneously. In this paper we propose a two-step method based on an attention model to impute missing values in multivariate energy time series. First, the underlying distribution of the missing values in the data is learned. This information is then further used to train an attention based imputation model. By learning the distribution prior to the imputation process, the model can respond flexibly to the specific characteristics of the underlying data. The developed model is applied to European energy data, obtained from the European Network of Transmission System Operators for Electricity. Using different evaluation metrics and benchmarks, the conducted experiments show that the proposed model is preferable to the benchmarks and is able to accurately impute missing values.

   @article{bulteMultivariateTimeSeries2023,
     title = {Multivariate Time Series Imputation for Energy Data Using Neural Networks},
     author = {Bülte, Christopher and Kleinebrahm, Max and Yilmaz, Hasan Ümitcan and Gómez-Romero, Juan},
     year = {2023},
     journal = {Energy and AI},
     volume = {13},
     pages = {100239},
     issn = {2666-5468},
     doi = {10.1016/j.egyai.2023.100239},
     url = {https://www.sciencedirect.com/science/article/pii/S2666546823000113},
     keywords = {Attention model, Energy data, Missing value estimation, Multivariate time series, Neural networks},
   }