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.
To demonstrate the effectiveness of B2B operator learning, we present a qualitative example on an elasticity dataset. The dataset consists of a plate with a circular cutout. A forcing function is applied, and the goal is to predict the displacement of the plate. B2B can accurately predict the displacement of the plate from data on the forcing function alone.
Interestingly, decomposing the operator learning problem into separate parts leads to smoother loss
landscapes compared to DeepONet, which leverages an end-to-end approach.
This likely explains the faster convergence and better asymptotic performance of B2B.
B2B is rooted in classical functional analysis. As a result, with minimal modifications, B2B can recover the algorithms for eigen decomposition and singular value decomposition.
While these approaches can only learn linear operators, they provide a means for analysing learned operators.
@misc{basistobasisoperatorlearning,
title={Basis-to-Basis Operator Learning Using Function Encoders},
author={Tyler Ingebrand and Adam J. Thorpe and Somdatta Goswami and Krishna Kumar and Ufuk Topcu},
year={2024},
eprint={2410.00171},
archivePrefix={arXiv},
url={https://arxiv.org/abs/2410.00171},
}