Christopher E. Miles

Ray Zirui Zhang, Christopher E. Miles, Xiaohui Xie et al.

We propose a new neural network based method for solving inverse problems for partial differential equations (PDEs) by formulating the PDE inverse problem as a bilevel optimization problem. At the upper level, we minimize the data loss with respect to the PDE parameters. At the lower level, we train a neural network to locally approximate the PDE solution operator in the neighborhood of a given set of PDE parameters, which enables an accurate approximation of the descent direction for the upper level optimization problem. The lower level loss function includes the L2 norms of both the residual and its derivative with respect to the PDE parameters. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. The method, which we refer to as BiLO (Bilevel Local Operator learning), is also able to efficiently infer unknown functions in the PDEs through the introduction of an auxiliary variable. We provide a theoretical analysis that justifies our approach. Through extensive experiments over multiple PDE systems, we demonstrate that our method enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need to balance the residual and the data loss, which is inherent to the soft PDE constraints in many existing methods.

Ray Zirui Zhang, Christopher E. Miles, Xiaohui Xie et al.

Uncertainty quantification and inverse problems governed by partial differential equations (PDEs) are central to a wide range of scientific and engineering applications. In this second part of a two part series, we extend Bilevel Local Operator Learning (BiLO) for PDE-constrained optimization problems developed in Part 1 to the Bayesian inference framework. At the lower level, we train a network to approximate the local solution operator by minimizing the local operator loss with respect to the weights of the neural network. At the upper level, we sample the PDE parameters from the posterior distribution. We achieve efficient sampling through gradient-based Markov Chain Monte Carlo (MCMC) methods and low-rank adaptation (LoRA). Compared with existing methods based on Bayesian neural networks, our approach bypasses the challenge of sampling in the high-dimensional space of neural network weights and does not require specifying a prior distribution on the neural network solution. Instead, uncertainty propagates naturally from the data through the PDE constraints. By enforcing strong PDE constraints, the proposed method improves the accuracy of both parameter inference and uncertainty quantification. We analyze the dynamic error of the gradient in the MCMC sampler and the static error in the posterior distribution due to inexact minimization of the lower level problem and demonstrate a direct link between the tolerance for solving the lower level problem and the accuracy of the resulting uncertainty quantification. Through numerical experiments across a variety of PDE models, we demonstrate that our method delivers accurate inference and quantification of uncertainties while maintaining high computational efficiency.