Generative Priors Conditioned on Natural Language for Bayesian Inversion in PDEs

Pengyu Zhang, Mark Girolami, Arnaud Vadeboncoeur

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…