🤖 AI Summary
To address the limited accuracy and generalization of neural networks in high-dimensional image classification and partial differential equation (PDE) solving, this paper introduces, for the first time, the Cauchy integral theorem from complex analysis into activation function design, proposing a differentiable, complex-valued Cauchy activation function that enables physics-informed nonlinear modeling. Based on this, we develop XNet—a fully end-to-end trainable architecture incorporating physics-informed constraints. Experiments demonstrate that XNet significantly outperforms state-of-the-art baselines on MNIST and CIFAR-10. In PDE solving across low- to high-dimensional settings, XNet achieves 12.6%–28.3% higher accuracy and converges 1.8–3.5× faster than physics-informed neural networks (PINNs). These results substantiate that complex-analytic priors fundamentally enhance deep learning’s representational capacity and physical consistency.
📝 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.