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.

中文翻译：

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使用函数编码器进行基础到基础的操作员学习


我们提出了 Basis-to-Basis （B2B） 算子学习，这是一种基于函数编码器的基本思想，用于学习函数的 Hilbert 空间算子的新方法。我们将学习算子的任务分解为两部分：学习输入和输出空间的基函数集，以及学习基函数系数之间的潜在非线性映射。B2B 操作员学习通过利用经典技术（如最小二乘法）来计算系数，从而规避了先前工作的许多挑战，例如要求数据位于固定位置。它对于线性算子特别有效，在线性算子中，我们将基之间的映射计算为具有封闭式解的单个矩阵变换。此外，通过最少的修改并利用函数编码器和函数分析之间的深层理论联系，我们推导出了直接类似于特征分解和奇异值分解的算子学习算法。我们在 7 个基准运算符学习任务上实证验证了 B2B 运算符学习，并表明它在几个基准任务上的准确性比现有方法提高了两个数量级。