The fundamental computational issues in Bayesian inverse problems (BIPs) governed by partial differential equations (PDEs) stem from the requirement of repeated forward model evaluations. A popular strategy to reduce such cost is to replace expensive model simulations by computationally efficient approximations using operator learning, motivated by recent progresses 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 fine-tuning the pre-trained approximate model during the inversion process with adaptive points selected by a greedy algorithm, which requires only a few forward model evaluations. To validate our approach, we adopt DeepOnet to construct the surrogate and use unscented Kalman inversion (UKI) to approximate the solution of BIPs, respectively. Furthermore, we present rigorous convergence guarantee in the linear case using the framework of UKI. We test the approach on several benchmarks, including the Darcy flow, the heat source inversion problem, and the reaction diffusion problems. Numerical results demonstrate that our method can significantly reduce computational costs while maintaining inversion accuracy.

翻译：由偏微分方程控制的贝叶斯反问题中的基本计算难题源于对正向模型重复评估的需求。受深度学习最新进展推动，一种降低此类成本的流行策略是利用算子学习构建计算高效的近似模型，以替代昂贵的模型模拟。然而，直接使用近似模型可能引入建模误差，加剧反问题本身存在的不适定性。因此，在精度与效率之间取得平衡对于此类方法的有效实施至关重要。为此，我们开发了一种自适应算子学习框架，该框架通过迫使代理模型在局部区域保持精确来逐步降低建模误差。这通过在反演过程中利用贪婪算法选取自适应点，对预训练的近似模型进行微调实现，仅需少量正向模型评估。为验证所提方法，我们分别采用DeepOnet构建代理模型，并利用无迹卡尔曼反演近似求解贝叶斯反问题。此外，我们在线性情形下基于无迹卡尔曼反演框架给出了严格的收敛性保证。我们在多个基准算例上测试了该方法，包括达西流、热源反演问题及反应扩散问题。数值结果表明，我们的方法能在保持反演精度的同时显著降低计算成本。