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 seven benchmark operator learning tasks and show that it demonstrates a two-orders-of-magnitude improvement in accuracy over existing approaches on several benchmark tasks.

翻译：本文提出基到基（B2B）算子学习方法，这是一种基于函数编码器基本原理、在函数希尔伯特空间上学习算子的新方法。我们将算子学习任务分解为两部分：学习输入与输出空间的基函数集合，以及学习基函数系数之间的潜在非线性映射。B2B算子学习通过利用最小二乘法等经典技术计算系数，规避了先前工作中的诸多挑战（例如要求数据位于固定位置）。该方法对线性算子尤其有效，此时我们将基之间的映射计算为具有闭式解的单矩阵变换。此外，通过极小修改并利用函数编码器与泛函分析之间的深层理论关联，我们推导出直接类比于特征分解与奇异值分解的算子学习算法。我们在七个基准算子学习任务上实证验证了B2B算子学习方法，结果表明其在多个基准任务上的精度比现有方法提升两个数量级。