Zero-Shot Uncertainty Quantification using Diffusion Probabilistic Models

TL;DR

The paper tackles zero-shot uncertainty quantification for regression using diffusion probabilistic models by framing diffusion ensembles as Bayesian model averaging. It develops a MC-based diffusion-ensemble formulation to produce predictive distributions and demonstrates that ensemble means improve accuracy across 1D and 2D regression tasks while ensemble variance serves as a practical uncertainty proxy. Empirical results across PDE-Refiner, ACDM, and PI-DFS show consistent gains and a strong correlation between ensemble error and variance, enabling confidence monitoring without uncertainty-aware training. The work provides a simple, cost-aware strategy for balancing ensemble size against computational overhead, offering practitioners a straightforward tool for uncertainty quantification in diffusion-based regression surrogates.

Abstract

The success of diffusion probabilistic models in generative tasks, such as text-to-image generation, has motivated the exploration of their application to regression problems commonly encountered in scientific computing and various other domains. In this context, the use of diffusion regression models for ensemble prediction is becoming a practice with increasing popularity. Under such background, we conducted a study to quantitatively evaluate the effectiveness of ensemble methods on solving different regression problems using diffusion models. We consider the ensemble prediction of a diffusion model as a means for zero-shot uncertainty quantification, since the diffusion models in our study are not trained with a loss function containing any uncertainty estimation. Through extensive experiments on 1D and 2D data, we demonstrate that ensemble methods consistently improve model prediction accuracy across various regression tasks. Notably, we observed a larger accuracy gain in auto-regressive prediction compared with point-wise prediction, and that enhancements take place in both the mean-square error and the physics-informed loss. Additionally, we reveal a statistical correlation between ensemble prediction error and ensemble variance, offering insights into balancing computational complexity with prediction accuracy and monitoring prediction confidence in practical applications where the ground truth is unknown. Our study provides a comprehensive view of the utility of diffusion ensembles, serving as a useful reference for practitioners employing diffusion models in regression problem-solving.

Figures (8)

• Figure 1: A overview of uncertainty quantification using diffusion probabilistic ensemble. With $J$$(J\geq2)$ samples of model prediction, we can estimate the prediction uncertainty at a given spatial location (lower-left plot), enhance the prediction accuracy via ensembled prediction (lower-middle plot), and use the ensemble variance to evaluate the prediction confidence (lower-right plot, where regions in dark purple color indicate a lower prediction confidence).
• Figure 2: Dynamic Time Warping similarity between the spatial means of ensemble prediction variance and ensemble prediction error, where the Euclidean distance is used as the distance metric and the color map (varying from white to blue) for coloring the background.
• Figure 3: The plots of mean ensemble variances for different ensemble sizes. The ensemble size are chosen as 7 by observing the convergence of the mean variance with respect to the increase of ensemble size to balance the benefit of ensemble and its computational complexity.
• Figure 4: A comparison of prediction samples and their ensemble from PDE-Refiner.
• Figure 5: A comparison of prediction samples and their ensemble from ACDM.