  We propose an operator learning approach to accelerate geometric Markov chain
Monte Carlo (MCMC) for solving infinite-dimensional Bayesian inverse problems
(BIPs). While geometric MCMC employs high-quality proposals that adapt to
posterior local geometry, it requires repeated computations of gradients and
Hessians of the log-likelihood, which becomes prohibitive when the
parameter-to-observable (PtO) map is defined through expensive-to-solve
parametric partial differential equations (PDEs). We consider a
delayed-acceptance geometric MCMC method driven by a neural operator surrogate
of the PtO map, where the proposal exploits fast surrogate predictions of the
log-likelihood and, simultaneously, its gradient and Hessian. To achieve a
substantial speedup, the surrogate must accurately approximate the PtO map and
its Jacobian, which often demands a prohibitively large number of PtO map
samples via conventional operator learning methods. In this work, we present an
extension of derivative-informed operator learning [O'Leary-Roseberry et al.,
J. Comput. Phys., 496 (2024)] that uses joint samples of the PtO map and its
Jacobian. This leads to derivative-informed neural operator (DINO) surrogates
that accurately predict the observables and posterior local geometry at a
significantly lower training cost than conventional methods. Cost and error
analysis for reduced basis DINO surrogates are provided. Numerical studies
demonstrate that DINO-driven MCMC generates effective posterior samples 3--9
times faster than geometric MCMC and 60--97 times faster than prior
geometry-based MCMC. Furthermore, the training cost of DINO surrogates breaks
even compared to geometric MCMC after just 10--25 effective posterior samples.
