There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint. In this paper, we partially explain this mystery by proving that without querying the adjoint, one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis. We then apply the result to recovering Green's functions of elliptic partial differential operators and derive an adjoint-free sample complexity bound. While existing theory justifies low sample complexity in operator learning, ours is the first adjoint-free analysis that attempts to close the gap between theory and practice.

翻译：算子学习领域存在一个核心谜题：如何在不探测伴随算子的情况下从数据中恢复非自伴算子？当前实践方法表明，仅利用算子正向作用生成的数据（无需访问伴随算子）即可实现算子的精确恢复。然而直观而言，对伴随算子作用的采样似乎是必要的。本文通过证明无需查询伴随算子即可通过傅里叶基投影逼近一类非自伴无限维紧致算子，部分解释了这一谜题。我们进一步将该结果应用于椭圆型偏微分算子格林函数的恢复，推导出无需伴随算子的样本复杂度界。现有理论虽已论证算子学习的低样本复杂度特性，但本文首次提出了一种旨在弥合理论与实际差距的无伴随分析方法。