Ray Zirui Zhang (WPI): Bilevel Local Operator Learning (BiLO) for PDE inverse problems

Many scientific models are formulated as partial differential equations (PDEs) describing physical or biological processes. In practice, key parameters or functions in these models are often unknown and must be inferred from limited and noisy data, leading to PDE inverse problems.

 

In this talk, I present Bilevel Local Operator Learning (BiLO), a neural-network–based framework for solving PDE inverse problems: at the lower level, a neural network learns a local approximation of the PDE solution operator, enabling accurate updates of the unknown parameters and functions at the upper level. This bilevel formulation avoiding the delicate trade-off between fitting data and satisfying the model common in many existing methods. I will also present a Bayesian extension, Bayesian BiLO, for uncertainty quantification. By combining Hamiltonian Monte Carlo (HMC) with lightweight neural-network updates via low-rank adaptation (LoRA), the method enables efficient gradient-based sampling. Examples include inferring parameters governing tumor growth and infiltration from medical imaging data.