The present paper proposes a Bayesian framework for inverse problems that seamlessly integrates optimization and inversion to enable rapid surrogate modeling, accurate parameter inference, and rigorous uncertainty quantification. Bayesian optimization is employed to adaptively construct accurate Gaussian process surrogate models using a minimal number of high-fidelity model evaluations, strategically focusing sampling in regions of high predictive uncertainty. The trained surrogate model is then leveraged within a Bayesian inversion scheme to infer optimal parameter values by combining prior knowledge with observed quantities of interest, resulting in posterior distributions that rigorously characterize epistemic uncertainty. The framework is theoretically grounded, computationally efficient, and particularly suited for engineering applications in which high-fidelity models -- whether arising from numerical simulations or physical experiments -- are computationally expensive, analytically intractable, or difficult to replicate, and data availability is limited. Furthermore, the combined use of Bayesian optimization and inversion outperforms their separate application, highlighting the synergistic benefits of unifying the two approaches. The performance of the proposed Bayesian framework is demonstrated on a suite of one- and two-dimensional analytical benchmarks, including the Mixed Gaussian-Periodic, Lévy, Griewank, Forrester, and Rosenbrock functions, which provide a controlled setting to assess surrogate modeling accuracy, parameter inference robustness, and uncertainty quantification. The results demonstrate the framework's effectiveness in efficiently solving inverse problems while providing informative uncertainty quantification and supporting reliable engineering decision-making at reduced computational cost.

翻译：本文提出一种贝叶斯逆问题求解框架，通过优化与反演的无缝集成实现快速代理建模、精确参数推断与严格不确定性量化。该框架采用贝叶斯优化方法，以最少的高保真模型评估次数自适应构建精确的高斯过程代理模型，并通过在预测不确定性较高区域进行策略性采样来提升建模效率。训练完成的代理模型随后被嵌入贝叶斯反演架构中，通过融合先验知识与观测目标量来推断最优参数值，最终生成严格表征认知不确定性的后验分布。本框架具有理论严谨性、计算高效性等特点，特别适用于高保真模型（无论是数值模拟还是物理实验产生）计算成本高昂、解析处理困难或难以复现，且数据获取受限的工程应用场景。此外，贝叶斯优化与反演的联合应用相较于各自独立实施展现出更优性能，凸显了两种方法协同整合的优势。通过在一维和二维解析基准测试集（包括混合高斯-周期函数、Lévy函数、Griewank函数、Forrester函数和Rosenbrock函数）上的验证，该框架在代理建模精度、参数推断鲁棒性和不确定性量化等方面得到系统评估。实验结果表明，所提框架能够以较低计算成本有效求解逆问题，同时提供信息丰富的不确定性量化结果，为可靠的工程决策提供支撑。