Zero-Shot Uncertainty Quantification using Diffusion Probabilistic Models

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.

• The paper introduces a Bayesian formulation for diffusion model ensembles to perform zero-shot uncertainty quantification in regression tasks.
• It demonstrates through experiments that ensemble variance strongly correlates with prediction errors, validating uncertainty estimates.
• It offers practical guidance on selecting ensemble size to balance computational cost with improved prediction accuracy across models.

Zero-Shot Uncertainty Quantification using Diffusion Probabilistic Models

The paper "Zero-Shot Uncertainty Quantification using Diffusion Probabilistic Models" by Dule Shu and Amir Barati Farimani introduces an innovative approach to zero-shot uncertainty quantification in regression tasks using diffusion probabilistic models. This research examines the potential of diffusion models for ensemble predictions, specifically for generating probabilistic outputs, without incorporating an uncertainty estimation loss function during training. The efficacy of this methodology is demonstrated across various regression problems, highlighting significant improvements in prediction accuracy and providing a straightforward and useful tool for uncertainty quantification (UQ).

Key Contributions

1. Bayesian Formulation for Diffusion Ensembles: The authors extend the traditional Deep Ensembles method into the domain of diffusion probabilistic models, offering a Bayesian perspective using Bayesian Model Averaging (BMA). This formulation outlines how the inherent sampling variability in diffusion processes can be leveraged for ensemble predictions and uncertainty estimation.
2. Ensemble Experiments Across Diffusion Models: The study conducts comprehensive experiments using three distinct diffusion models: PDE-Refiner, ACDM, and PI-DFS, evaluating their performance on different regression tasks. The novelty here lies in pioneering evaluation of ensemble effectiveness across various designs of diffusion regression models.
3. Correlation Between Prediction Error and Ensemble Variance: Through rigorous numerical experiments, it’s demonstrated that the ensemble variance correlates strongly with prediction errors, validating ensemble variance as a reliable indicator of prediction uncertainty.
4. Analysis of Ensemble Size: By identifying the relationship between ensemble size and computational cost, a practical method for selecting an appropriate ensemble size to balance performance and efficiency is proposed.

Experimental Insights

Improvements in Accuracy

The paper presents robust evidence that ensemble methods increase prediction accuracy across evaluated models and tasks. For instance, in the case of PDE-Refiner, an auto-regressive prediction model for 1D data, ensemble predictions achieved a relative $\mathcal{L}^2$ error reduction from 0.1192 to 0.0986. Similarly, ACDM, used for 2D data, showed a reduction in relative $\mathcal{L}^2$ error from 0.5516 to 0.3966. PI-DFS, designed for point-wise prediction in 2D data, not only demonstrated improvements in $\mathcal{L}^2$ error but also in physics-informed residual loss, with the latter decreasing from 0.2586 to 0.1846 for ensemble predictions.

Correlation with Ensemble Variance

A significant observation across models is the high Pearson correlation between prediction errors and ensemble variance, with values close to 1. This correlation suggests that ensemble variance can serve as a proxy for assessing prediction reliability, offering a valuable metric where ground truth labels are unavailable.

Practical Implementation

An essential aspect of the research focuses on the computational complexity of ensemble methods. By analyzing the convergence of ensemble variance with respect to ensemble size, the study provides practical insights for optimizing computational resources. For example, the optimal ensemble size is empirically established to balance the computational overhead against the predictive accuracy gains, concluding an optimal size at approximately 7 for the models evaluated.

Implications and Future Work

The practicality of this research lies in its ability to seamlessly integrate uncertainty quantification into regression tasks using diffusion probabilistic models. This methodology facilitates more reliable predictions in scientific computing and various applied domains without the need for additional uncertainty estimation during training.

Future developments could explore the intricacies of the observed correlation between ensemble error and variance further. Potential advancements could include importance sampling strategies based on ensemble uncertainty to enhance the efficiency of training data usage and model fine-tuning. By refining these methods, the predictive power and applicability of diffusion models in regression tasks could be significantly expanded, paving the way for more precise and reliably quantifiable models in machine learning and scientific computing.

By integrating these insights, the approach proposed in this paper stands as a comprehensive reference for leveraging diffusion probabilistic models in regression tasks, shedding light on both theoretical frameworks and practical implementations.