Neural Implementations of Rational Approximation: CauchyNet and XNet

Xin Li, Xiaotao Zheng, Zhihong Jeff Xia, Hong-Kun Zhang

Keywords: Rational Approximation, Complex Analysis, Function Approximation, Partial Differential Equations

TL;DR: We introduce CauchyNet and XNet, neural architectures leveraging Cauchy integral formula for scalable rational function approximation in solving PDEs.

Abstract:

Rational approximants often outperform polynomials, especially near nonsmooth structure. On bounded 1D domains, they attain optimal rates (exponential for analytic targets; root-exponential for analytic functions with finitely many singularities). Yet scalable neural parameterizations with classical rates are limited.
We propose \textbf{CauchyNet}, a rational parameterization from the Cauchy integral formula that we implemented in a neural network.
For scalability, we employ \textbf{XNet}, a ridge-projected Cauchy layer with linear $\mathcal{O}(MN)$ complexity.

Across parameter-matched approximation and PDE tests, Cauchy-based models show the expected rate diagnostics and strong accuracy–compute trade-offs.

Primary Area: learning theory

Submission Number: 24563