Thomas O'Leary-Roseberry
University of Texas Austin
Title
Efficient Infinite-Dimensional Bayesian Inversion using Derivative-Informed Neural Operators
Abstract
We address Bayesian inverse problems (BIPs) for functions that parametrize PDE models, such as heterogeneous coefficient fields. These problems are challenging due to high-dimensional parameter spaces upon discretization, the computational demands of PDE-based likelihood evaluations, and the complex geometry of the posterior (e.g., concentration and non-linear multi-modality). While efficient algorithms for infinite-dimensional BIPs leverage posterior geometry through likelihood derivatives, they become impractical when the computational model (and its adjoints) is expensive to compute. In this talk, we introduce derivative-informed neural operators (DINOs), a class of neural operator surrogates trained to accurately approximate both PDE mappings and their derivatives (Fréchet derivatives). DINOs offer superior generalization over traditional neural operators, enabling precise derivative use in high-dimensional uncertainty quantification (UQ) tasks like BIPs. Specifically, reduced basis DINOs are a natural choice for algorithms that employ subspace decompositions. In the context of BIPs, these algorithms allowing for efficient inversion in a likelihood-dominated subspace, with the prior completing the orthogonal component. We will explore these algorithms within the frameworks of geometric MCMC and transport map variational inference. Our numerical results show that DINO-accelerated algorithms achieve state-of-the-art cost-accuracy performance compared to various other methods.
