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