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Peer-reviewedCC BYNature communications

Bayesian bilevel operator learning with low-rank adaptation for efficient uncertainty quantification of PDE inverse problems.

  • Ray Zirui Zhang1
  • Christopher E Miles2
  • Xiaohui Xie3
  • John S Lowengrub4,5
5 affiliations
  1. 1Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, USA. [email protected].
  2. 2Department of Mathematics, University of California, Irvine, Irvine, CA, USA.
  3. 3Department of Computer Science, University of California, Irvine, Irvine, CA, USA.
  4. 4Department of Mathematics, University of California, Irvine, Irvine, CA, USA. [email protected].
  5. 5Department of Biomedical Engineering, University of California, Irvine, Irvine, CA, USA. [email protected].
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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.

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Cite

BibTeXRIS
BibTeX
@article{zhang2026bayesian,
  title = {{Bayesian bilevel operator learning with low-rank adaptation for efficient uncertainty quantification of PDE inverse problems.}},
  author = {Ray Zirui Zhang and Christopher E Miles and Xiaohui Xie and John S Lowengrub},
  journal = {Nature communications},
  year = {2026},
  month = sep,
  doi = {10.1038/s41467-026-77768-7},
  url = {https://europepmc.org/article/MED/42901898},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Bayesian bilevel operator learning with low-rank adaptation for efficient uncertainty quantification of PDE inverse problems.
AU  - Ray Zirui Zhang
AU  - Christopher E Miles
AU  - Xiaohui Xie
AU  - John S Lowengrub
PY  - 2026
DA  - 2026/09/19
JO  - Nature communications
DO  - 10.1038/s41467-026-77768-7
UR  - https://europepmc.org/article/MED/42901898
AB  - 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.
SN  - 2041-1723
ER  -