Generative Priors Conditioned on Natural Language for Bayesian Inversion in PDEs

2026-09-26

AI Summary

This study addresses the challenge of inferring physical quantities by fusing qualitative textual descriptions with quantitative data in partial differential equation (PDE) inverse problems. To this end, we propose a natural language-driven generative Bayesian framework. This method pioneers the incorporation of natural language as conditional constraints into Bayesian prior construction, enabling uncertainty quantification from sparse noisy observations. Furthermore, it leverages conditional diffusion models and autoencoders to perform high-dimensional posterior sampling, substantially improving multimodal fusion efficiency. Experimental evaluations on heat equation benchmarks and meteorological datasets demonstrate that the proposed approach outperforms conventional baselines in both inference accuracy and uncertainty calibration.

Abstract

Inferring quantities of interest (QoI) from data is a central task in Science and Engineering. In such contexts, we often have access to both quantitative data and qualitative data. Quantitative data may be represented by noisy sensor measurements, simulation data, re-analysis data; qualitative data may be in the form of text descriptions of experimental setups, expected experiment outcomes, and human-perceived system behaviours. The task we address in this paper is the following. Given a training set of paired qualitative text and quantitative QoI data, we learn to exploit the inherent correlation between the two modalities to learn a highly informative data-driven natural-language-conditional Bayesian prior, such that when presented with a new physical system, we can coherently combine (i) the training dataset, (ii) qualitative text describing the new system, and (iii) a small number of noisy sensor readings from that new system, to perform inference and uncertainty quantification (UQ) over the QoI. To achieve this task, we develop two parallel approaches, one uses conditional diffusion and the other conditional autoencoders, and compare both against classical Bayesian methodology, unconditional generative models and deterministic supervised methods. Each approach has specific strengths and tradeoffs; conditional autoencoder offers theoretical tractability, allows for fast posterior sampling, and provides better-calibrated UQ, whereas conditional diffusion is explored for greater expressiveness and capturing complex posteriors with irregular QoI fields. The approach is tested on the steady-state heat equation, damped Helmholtz equation, and UK weather reanalysis data.

Problem

• Bayesian inversion
• generative priors
• natural language conditioning
• uncertainty quantification
• PDEs

Innovation

• Bayesian Inversion
• Conditional Generative Priors
• Natural Language Conditioning
• Diffusion Models
• Uncertainty Quantification