Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators. This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. We explore this for widely-used operator learning techniques. Our findings detail how aliasing introduces errors when handling different discretizations and grids and loss of crucial continuous structures. More generally, this framework not only sheds light on existing challenges but, given its constructive and broad nature, also potentially offers tools for developing new neural operators.

翻译：摘要：近年来，算子学习（即学习无限维函数空间之间的映射）引起了广泛关注，尤其是在从数据中学习偏微分方程方面。虽然在理论上概念清晰，但神经算子在转向计算机实现时需要离散化。这一步骤可能会损害其完整性，常常导致它们偏离底层算子。本研究提出了一种新的神经算子视角，即表示等价神经算子框架，旨在解决这些问题。其核心是算子混叠概念，用于衡量神经算子与其离散表示之间的不一致性。我们针对广泛使用的算子学习技术进行了探索。研究结果详细说明了当处理不同离散化和网格时，混叠如何引入误差，并导致关键连续结构的丢失。更广泛地说，该框架不仅揭示了现有挑战，而且由于其建设性和通用性，还可能为开发新的神经算子提供工具。