Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and differentiable map between the parametric input and the branch-net output. This allows us to construct accurate approximations of the full posterior, irrespective of the number of observations and the magnitude of the observation noise, without any need for additional forward solves nor for cumbersome, iterative sampling procedures. We demonstrate the efficacy and accuracy of the proposed methodology in the context of inverse problems for three benchmarks: an anti-derivative equation, reaction-diffusion dynamics and flow through porous media.

翻译：神经算子为求解参数化偏微分方程提供了强大的数据驱动工具，因其能够表示无穷维函数空间之间的映射。本文在贝叶斯高维逆问题背景下采用物理信息驱动的神经算子。传统求解策略需进行大量（且常不可行）的正向模型求解，同时还需计算参数导数。为实现高效求解，我们扩展深度算子网络（DeepONets），采用RealNVP架构构建参数输入与分支网络输出之间的可逆可微映射。该设计使得我们能够在无需额外正向求解或繁琐迭代采样过程的情况下，构建完整后验分布的精确近似，且该近似结果不受观测数据数量及观测噪声幅度的影响。通过三个基准逆问题（反导数方程、反应扩散动力学及多孔介质流动）验证了所提方法的有效性与精确性。