A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

翻译：一大类PDE反问题仅被定义为从算子到函数的映射。现有的算子学习框架执行的是函数到函数的映射，需经修改才能学习由数据驱动的逆映射。我们提出一种名为神经逆算子（NIOs）的新型架构来解决这些PDE反问题。受底层数学结构启发，NIO基于DeepONet和FNO的合理组合来逼近从算子到函数的映射。多项实验表明，NIO显著优于基线方法，能够稳健、精确地求解PDE反问题，且其求解速度比现有的直接优化方法和PDE约束优化方法快数个数量级。