Basis-to-Basis Operator Learning Using Function Encoders

TL;DR

This paper introduces Basis-to-Basis (B2B) operator learning, a new method that decomposes operator learning into basis function learning and coefficient mapping, achieving significant accuracy improvements on benchmark tasks.

Contribution

The paper proposes B2B operator learning, a novel approach that leverages function encoders and classical techniques to improve operator learning accuracy and flexibility.

Findings

Achieves two-orders-of-magnitude accuracy improvement on benchmarks

Effectively handles linear and nonlinear operators with basis decomposition

Demonstrates robustness and efficiency across seven benchmark tasks

Abstract

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.