Sequential Bayesian Design for Efficient Surrogate Construction in the Inversion of Darcy Flows
This work addresses the computational bottleneck in Bayesian inversion for PDE-based problems, offering a domain-specific incremental improvement for fields like fluid mechanics and engineering.
The study tackled the challenge of constructing efficient surrogate models for high-dimensional inverse problems governed by PDEs, such as Darcy flow, by proposing a locally accurate surrogate that focuses on high-probability regions, resulting in improved inversion accuracy and computational speed as demonstrated in three experiments.
Inverse problems governed by partial differential equations (PDEs) play a crucial role in various fields, including computational science, image processing, and engineering. Particularly, Darcy flow equation is a fundamental equation in fluid mechanics, which plays a crucial role in understanding fluid flow through porous media. Bayesian methods provide an effective approach for solving PDEs inverse problems, while their numerical implementation requires numerous evaluations of computationally expensive forward solvers. Therefore, the adoption of surrogate models with lower computational costs is essential. However, constructing a globally accurate surrogate model for high-dimensional complex problems demands high model capacity and large amounts of data. To address this challenge, this study proposes an efficient locally accurate surrogate that focuses on the high-probability regions of the true likelihood in inverse problems, with relatively low model complexity and few training data requirements. Additionally, we introduce a sequential Bayesian design strategy to acquire the proposed surrogate since the high-probability region of the likelihood is unknown. The strategy treats the posterior evolution process of sequential Bayesian design as a Gaussian process, enabling algorithmic acceleration through one-step ahead prior. The complete algorithmic framework is referred to as Sequential Bayesian design for locally accurate surrogate (SBD-LAS). Finally, three experiments based the Darcy flow equation demonstrate the advantages of the proposed method in terms of both inversion accuracy and computational speed.