Cauchy activation function and XNet

TL;DR

This paper introduces the Cauchy Activation Function and the novel XNet architecture, demonstrating superior performance in high-dimensional image classification and PDE solving compared to existing methods.

Contribution

The paper presents a new activation function based on complex analysis and a neural network architecture tailored for high-precision, high-dimensional problems.

Findings

XNet outperforms benchmarks like MNIST and CIFAR-10 in image classification.

XNet shows significant advantages over PINNs in PDE tasks.

The Cauchy Activation Function enhances neural network precision for complex problems.

Abstract

We have developed a novel activation function, named the Cauchy Activation Function. This function is derived from the Cauchy Integral Theorem in complex analysis and is specifically tailored for problems requiring high precision. This innovation has led to the creation of a new class of neural networks, which we call (Comple)XNet, or simply XNet. We will demonstrate that XNet is particularly effective for high-dimensional challenges such as image classification and solving Partial Differential Equations (PDEs). Our evaluations show that XNet significantly outperforms established benchmarks like MNIST and CIFAR-10 in computer vision, and offers substantial advantages over Physics-Informed Neural Networks (PINNs) in both low-dimensional and high-dimensional PDE scenarios.