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Towards Intrinsically Calibrated Uncertainty Quantification in Industrial Data-Driven Models via Diffusion Sampler

Yiran Ma, Jerome Le Ny, Zhichao Chen, Zhihuan Song

Abstract

In modern process industries, data-driven models are important tools for real-time monitoring when key performance indicators are difficult to measure directly. While accurate predictions are essential, reliable uncertainty quantification (UQ) is equally critical for safety, reliability, and decision-making, but remains a major challenge in current data-driven approaches. In this work, we introduce a diffusion-based posterior sampling framework that inherently produces well-calibrated predictive uncertainty via faithful posterior sampling, eliminating the need for post-hoc calibration. In extensive evaluations on synthetic distributions, the Raman-based phenylacetic acid soft sensor benchmark, and a real ammonia synthesis case study, our method achieves practical improvements over existing UQ techniques in both uncertainty calibration and predictive accuracy. These results highlight diffusion samplers as a principled and scalable paradigm for advancing uncertainty-aware modeling in industrial applications.

Towards Intrinsically Calibrated Uncertainty Quantification in Industrial Data-Driven Models via Diffusion Sampler

Abstract

In modern process industries, data-driven models are important tools for real-time monitoring when key performance indicators are difficult to measure directly. While accurate predictions are essential, reliable uncertainty quantification (UQ) is equally critical for safety, reliability, and decision-making, but remains a major challenge in current data-driven approaches. In this work, we introduce a diffusion-based posterior sampling framework that inherently produces well-calibrated predictive uncertainty via faithful posterior sampling, eliminating the need for post-hoc calibration. In extensive evaluations on synthetic distributions, the Raman-based phenylacetic acid soft sensor benchmark, and a real ammonia synthesis case study, our method achieves practical improvements over existing UQ techniques in both uncertainty calibration and predictive accuracy. These results highlight diffusion samplers as a principled and scalable paradigm for advancing uncertainty-aware modeling in industrial applications.

Paper Structure

This paper contains 30 sections, 3 theorems, 24 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uncertainty Quantification in Machine Learning
  4. Bayesian Learning
  5. Connections Between Sampling and Finite-Horizon Stochastic Optimal Control
  6. Uncertainty Quantification in Industrial Models via Diffusion Sampler
  7. Probabilistic Regression Model
  8. Posterior Sampling via Diffusion Sampler
  9. Training
  10. Sampling
  11. Posterior Prediction
  12. Experiments
  13. Evaluation Setup
  14. Metrics
  15. Baseline Methods
  16. ...and 15 more sections

Key Result

Lemma 1

Let $\mathbb{Q}, \mathbb{S}$ be two probability measures on path space, with terminal distributions $\mu_1 = \mathbb{Q}_1$, $\pi_1 = \mathbb{S}_1$. Then where the equality holds if and only if $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure 1: Overview of the DiffUQ Framework
  • Figure 2: Comparison on the smiley-face (left) and funnel (right) distributions. (a,e) show mean-field variational contours; other panels show samples from different sampling methods.
  • Figure 3: Summary of the first principle-based mathematical simulator of industrial-scale penicillin simulation goldrick2019modern
  • Figure 4: High-Low Transformer unit from an ammonia synthesis process.
  • Figure 5: Sensitivity Analysis. The shaded area indicates $\pm 1$ standard deviation over 5 independent runs.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Definition 1: Calibration
  • Definition 2: Calibrated regression kuleshov2018accurate
  • Definition 3: Point Predictions and Credible Intervals for Regression
  • Definition 4: Schrödinger bridge problem leonard2013survey
  • Lemma 1: Data processing inequality (See Appendix A in leonard2013survey)
  • Proposition 1: Stochastic optimal control formulation of Schrödinger bridge problemleonard2013survey
  • Proposition 2: Stochastic optimal control cost with soft terminal penalty tzen2019theoreticalvargas2023bayesianzhang2021path