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.

翻译：我们提出一种算子学习方法，用于加速求解无限维贝叶斯反问题（BIPs）的几何马尔可夫链蒙特卡洛（MCMC）方法。尽管几何MCMC采用能适应后验局部几何的高质量建议分布，但其需要反复计算对数似然的梯度和海森矩阵，当参数到观测（PtO）映射通过求解计算代价高昂的参数化偏微分方程（PDE）定义时，这种计算变得不可行。我们考虑一种由PtO映射的神经算子代理驱动的延迟接受几何MCMC方法，其中建议分布利用代理模型对对数似然及其梯度和海森矩阵的快速预测。要实现显著加速，代理模型必须精确近似PtO映射及其雅可比矩阵，而传统的算子学习方法往往需要大量PtO映射样本才能达到这一精度。为此，我们扩展了导数信息增强的算子学习[O'Leary-Roseberry等，《计算物理杂志》，第496卷(2024)]，该方法使用PtO映射及其雅可比矩阵的联合样本。由此产生的导数信息增强神经算子（DINO）代理能够以显著低于传统方法的训练成本，精确预测观测量及后验局部几何结构。我们提供了简化基DINO代理的成本与误差分析。数值实验表明，DINO驱动的MCMC生成有效后验样本的速度比几何MCMC快3-9倍，比基于先验几何的MCMC快60-97倍。此外，仅需10-25个有效后验样本，DINO代理的训练成本即可与几何MCMC相平衡。