Sequential Bayesian Design for Efficient Surrogate Construction in the Inversion of Darcy Flows

2025-07-23

AI Summary

In Bayesian inference for Darcy flow inverse problems, repeated evaluations of expensive forward solvers severely hinder computational efficiency. To address this, we propose the SBD-LAS framework: a Sequential Bayesian Design (SBD)-based approach that dynamically focuses on high-likelihood regions during posterior evolution to construct lightweight, locally accurate Gaussian process surrogates. Crucially, we introduce a one-step lookahead prior to accelerate iterative updates, substantially reducing training data requirements and model complexity. Unlike conventional methods relying on global, high-capacity surrogates, SBD-LAS achieves an optimal trade-off between accuracy and efficiency. In three canonical Darcy flow benchmark experiments, SBD-LAS accelerates inversion by 2–5× over standard Bayesian approaches while delivering superior convergence accuracy and more stable posterior estimates. The framework demonstrates exceptional computational resource efficiency and generalization capability.

Technology Category

• Machine Learning: Bayesian Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

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.

Problem

• Efficient surrogate construction for Darcy flow inversion
• Reducing computational cost in PDE Bayesian inversion
• Sequential design for locally accurate surrogate models

Innovation

• Efficient locally accurate surrogate model
• Sequential Bayesian design strategy
• Gaussian process for posterior evolution