Parameter Inference and Uncertainty Quantification with Diffusion Models: Extending CDI to 2D Spatial Conditioning

2026-01-23

AI Summary

This work addresses the challenge of parameter identifiability and uncertainty quantification in scientific inverse problems involving two-dimensional spatial observations, such as convergent-beam electron diffraction (CBED) patterns. It extends the conditional diffusion inversion (CDI) framework to the 2D spatial domain for the first time, establishing a Bayesian posterior inference approach grounded in conditional diffusion models to enable probabilistic multi-parameter estimation directly from spatial data. Experimental results demonstrate that CDI effectively distinguishes between identifiable and ambiguous parameters: it yields sharp posterior distributions for well-identified parameters while preserving appropriate uncertainty for non-identifiable ones. This calibrated representation of uncertainty significantly outperforms conventional regression methods, which often mask uncertainty, thereby achieving reliable and well-calibrated uncertainty quantification.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

Uncertainty quantification is critical in scientific inverse problems to distinguish identifiable parameters from those that remain ambiguous given available measurements. The Conditional Diffusion Model-based Inverse Problem Solver (CDI) has previously demonstrated effective probabilistic inference for one-dimensional temporal signals, but its applicability to higher-dimensional spatial data remains unexplored. We extend CDI to two-dimensional spatial conditioning, enabling probabilistic parameter inference directly from spatial observations. We validate this extension on convergent beam electron diffraction (CBED) parameter inference - a challenging multi-parameter inverse problem in materials characterization where sample geometry, electronic structure, and thermal properties must be extracted from 2D diffraction patterns. Using simulated CBED data with ground-truth parameters, we demonstrate that CDI produces well-calibrated posterior distributions that accurately reflect measurement constraints: tight distributions for well-determined quantities and appropriately broad distributions for ambiguous parameters. In contrast, standard regression methods - while appearing accurate on aggregate metrics - mask this underlying uncertainty by predicting training set means for poorly constrained parameters. Our results confirm that CDI successfully extends from temporal to spatial domains, providing the genuine uncertainty information required for robust scientific inference.

Problem

• uncertainty quantification
• inverse problems
• parameter inference
• spatial data
• diffraction patterns

Innovation

• Conditional Diffusion Models
• Uncertainty Quantification
• Inverse Problems
• Spatial Conditioning
• Parameter Inference