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.

翻译：我们提出了一种名为柯西激活函数的新型激活函数。该函数源自复分析中的柯西积分定理，专门针对需要高精度求解的问题而设计。这一创新催生了一类新型神经网络，我们称之为（复）XNet，简称为XNet。我们将证明XNet在处理高维挑战（如图像分类和求解偏微分方程）时具有显著优势。评估结果表明，在计算机视觉任务中，XNet在MNIST和CIFAR-10等基准数据集上明显优于现有方法；在求解偏微分方程方面，无论低维或高维场景，XNet相较于物理信息神经网络（PINNs）均展现出显著性能提升。