We present Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions based on the foundational ideas of function encoders. We decompose the task of learning operators into two parts: learning sets of basis functions for both the input and output spaces, and learning a potentially nonlinear mapping between the coefficients of the basis functions. B2B operator learning circumvents many challenges of prior works, such as requiring data to be at fixed locations, by leveraging classic techniques such as least-squares to compute the coefficients. It is especially potent for linear operators, where we compute a mapping between bases as a single matrix transformation with a closed form solution. Furthermore, with minimal modifications and using the deep theoretical connections between function encoders and functional analysis, we derive operator learning algorithms that are directly analogous to eigen-decomposition and singular value decomposition. We empirically validate B2B operator learning on six benchmark operator learning tasks, and show that it demonstrates a two-orders-of-magnitude improvement in accuracy over existing approaches on several benchmark tasks.

本文提出了一种新的算子学习方法——基于基函数的算子学习（B2B），旨在解决高维空间中算子的学习问题。通过将学习任务分解为输入和输出空间基函数的学习以及基函数系数之间非线性映射的学习，B2B算法有效克服了数据固定位置等挑战。研究表明，该方法在多个基准任务中的准确性提升了两个数量级，展示了其显著的应用潜力。

基于基函数的算子学习利用函数编码器