Bayesian bilevel operator learning with low-rank adaptation for efficient uncertainty quantification of PDE inverse problems
TL;DR
A Bilevel Local Operator Learning framework for Bayesian inference in PDEs (B-BiLO), which enables efficient gradient-based sampling without synthetic data or adjoint equations and avoids sampling in high-dimensional weight space, as in Bayesian neural networks, by optimizing weights deterministically.
Abstract
Uncertainty quantification in PDE inverse problems is essential in many applications. Scientific machine learning and AI enable data-driven learning of model components while preserving physical structure, and provide the scalability and adaptability needed for emerging imaging technologies and clinical insights. We develop a Bilevel Local Operator Learning framework for Bayesian inference in PDEs (B-BiLO). At the upper level, we sample parameters from the posterior via Hamiltonian Monte Carlo, while at the lower level we fine-tune a neural network via low-rank adaptation (LoRA) to approximate the solution operator locally. B-BiLO enables efficient gradient-based sampling without synthetic data or adjoint equations and avoids sampling in high-dimensional weight space, as in Bayesian neural networks, by optimizing weights deterministically. We analyze errors from approximate lower-level optimization and establish their impact on posterior accuracy. Numerical experiments across PDE models, including tumor growth, demonstrate that B-BiLO achieves accurate and efficient uncertainty quantification.