Probabilistic neural operators for functional uncertainty quantification
Christopher Bülte, Philipp Scholl, Gitta Kutyniok
TL;DR
The paper introduces the probabilistic neural operator (PNO), a framework that extends neural operators to output distributions over function spaces and trains them with strictly proper scoring rules, notably the energy score. It provides a theoretical justification showing the energy score is strictly proper in separable Hilbert spaces and demonstrates improved calibration and uncertainty representation across PDE benchmarks and a real-world ERA5 temperature task, with two instantiations PNOD and PNOR offering different distributional characteristics. Empirically, PNO achieves well-calibrated predictive distributions and competitive mean accuracy, particularly in chaotic or long-horizon settings, while incurring higher computational costs. The work offers a versatile approach to functional uncertainty quantification in neural surrogate models for PDEs, enabling better uncertainty-aware predictions and extreme-event analyses in scientific applications.
Abstract
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.