The paper addresses Bayesian inferences in inverse problems with uncertainty quantification involving a computationally expensive forward map associated with solving a partial differential equations. To mitigate the computational cost, the paper proposes a new surrogate model informed by the physics of the problem, specifically when the forward map involves solving a linear elliptic partial differential equation. The study establishes the consistency of the posterior distribution for this surrogate model and demonstrates its effectiveness through numerical examples with synthetic data. The results indicate a substantial improvement in computational speed, reducing the processing time from several months with the exact forward map to a few minutes, while maintaining negligible loss of accuracy in the posterior distribution.

翻译：本文针对涉及偏微分方程求解且计算成本高昂的正向映射问题，提出一种基于贝叶斯推断的不确定性量化反演方法。为降低计算开销，本文提出一种融合问题物理信息的全新替代模型，尤其适用于正向映射涉及求解线性椭圆型偏微分方程的场景。研究建立了该替代模型后验分布的一致性理论，并通过合成数据的数值算例验证其有效性。结果表明，该方法在保持后验分布精度损失可忽略的前提下，实现了计算效率的显著提升——将原始精确正向映射需数月完成的计算任务缩短至数分钟。