Recently there has been great interest in operator learning, where networks learn operators between function spaces from an essentially infinite-dimensional perspective. In this work we present results for when the operators learned by these networks are injective and surjective. As a warmup, we combine prior work in both the finite-dimensional ReLU and operator learning setting by giving sharp conditions under which ReLU layers with linear neural operators are injective. We then consider the case the case when the activation function is pointwise bijective and obtain sufficient conditions for the layer to be injective. We remark that this question, while trivial in the finite-rank case, is subtler in the infinite-rank case and is proved using tools from Fredholm theory. Next, we prove that our supplied injective neural operators are universal approximators and that their implementation, with finite-rank neural networks, are still injective. This ensures that injectivity is not `lost' in the transcription from analytical operators to their finite-rank implementation with networks. Finally, we conclude with an increase in abstraction and consider general conditions when subnetworks, which may be many layers deep, are injective and surjective and provide an exact inversion from a `linearization.' This section uses general arguments from Fredholm theory and Leray-Schauder degree theory for non-linear integral equations to analyze the mapping properties of neural operators in function spaces. These results apply to subnetworks formed from the layers considered in this work, under natural conditions. We believe that our work has applications in Bayesian UQ where injectivity enables likelihood estimation and in inverse problems where surjectivity and injectivity corresponds to existence and uniqueness, respectively.

翻译：近年来，算子学习引起了广泛关注，即网络从本质无限维的角度学习函数空间之间的算子。本文给出了这些网络所学习算子为单射和满射的条件。作为铺垫，我们结合有限维ReLU和算子学习领域的先前工作，给出了线性神经算子的ReLU层为单射的严格条件。随后，我们考虑激活函数为逐点双射的情形，并得到了该层为单射的充分条件。值得注意的是，这一结论在有限秩情形下是平凡的，但在无限秩情形下更为微妙，需借助弗雷德霍姆理论进行证明。接着，我们证明了所构造的单射神经算子是通用逼近器，且其有限秩神经网络实现仍保持单射性。这确保了单射性在从解析算子到有限秩网络实现的转录过程中不会"丢失"。最后，我们提升抽象层次，考虑可能包含多个子层的子网络为单射和满射的一般条件，并给出了基于"线性化"的精确逆映射。本节利用弗雷德霍姆理论和非线性积分方程的勒雷-肖德尔度理论，分析了神经算子在函数空间中的映射性质。在自然条件下，这些结论适用于本文所考虑各层构成的子网络。我们认为该工作可应用于贝叶斯不确定性量化（单射性实现似然估计）及逆问题（满射性与单射性分别对应解的存在性与唯一性）。