The fundamental computational issues in Bayesian inverse problems (BIP) governed by partial differential equations (PDEs) stem from the requirement of repeated forward model evaluations. A popular strategy to reduce such costs is to replace expensive model simulations with computationally efficient approximations using operator learning, motivated by recent progress in deep learning. However, using the approximated model directly may introduce a modeling error, exacerbating the already ill-posedness of inverse problems. Thus, balancing between accuracy and efficiency is essential for the effective implementation of such approaches. To this end, we develop an adaptive operator learning framework that can reduce modeling error gradually by forcing the surrogate to be accurate in local areas. This is accomplished by adaptively fine-tuning the pre-trained approximate model with train- ing points chosen by a greedy algorithm during the posterior computational process. To validate our approach, we use DeepOnet to construct the surrogate and unscented Kalman inversion (UKI) to approximate the BIP solution, respectively. Furthermore, we present a rigorous convergence guarantee in the linear case using the UKI framework. The approach is tested on a number of benchmarks, including the Darcy flow, the heat source inversion problem, and the reaction-diffusion problem. The numerical results show that our method can significantly reduce computational costs while maintaining inversion accuracy.

翻译：贝叶斯反问题中由偏微分方程支配时的基本计算难题源于需要重复进行正演模型评估。受深度学习最新进展的启发，减少此类成本的一种常用策略是使用算子学习构建计算高效的近似模型替代昂贵数值模拟。然而，直接使用近似模型会引入建模误差，加剧反问题本已存在的病态性。因此，平衡精度与效率对此类方法的有效实施至关重要。为此，我们提出自适应算子学习框架，通过迫使代理模型在局部区域保持精度来逐步降低建模误差。该框架在贝叶斯后验计算过程中，采用贪心算法自适应选择训练点对预训练近似模型进行微调。我们以DeepOnet构建代理模型、无迹卡尔曼反演（UKI）近似贝叶斯反问题解为例验证该方法，并在线性情形下利用UKI框架给出严格的收敛性保证。通过达西流动、热源反演及反应扩散问题等基准测试，数值结果表明该方法能在保持反演精度的同时显著降低计算成本。