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：神经操作器提供了一种强大的数据驱动工具,用于解决能代表无限功能空间之间地图的参数PDE。在这项工作中,我们利用物理学知情神经操作器处理高维、拜耳斯反向问题。传统的解决方案战略需要大量且往往是不可行的远方模型解决方案,以及参数衍生物的计算。为了实现有效的解决方案,我们通过使用RealNVP结构扩展深操作器网络(DeepONets),该结构在参数输入和分支网络输出之间产生不可逆和不同的映射。这使我们能够建立全后方形的精确近似,而不论观测次数和观测噪音的大小,不需要额外的前方解决方案,也不需要繁琐的、迭接的取样程序。我们用三个基准,即反射方程式、反射动力和通过多孔媒体流动,展示了拟议方法的功效和准确性。</s>