This paper presents the double-activation neural network (DANN), a novel network architecture designed for solving parabolic equations with time delay. In DANN, each neuron is equipped with two activation functions to augment the network's nonlinear expressive capacity. Additionally, a new parameter is introduced for the construction of the quadratic terms in one of two activation functions, which further enhances the network's ability to capture complex nonlinear relationships. To address the issue of low fitting accuracy caused by the discontinuity of solution's derivative, a piecewise fitting approach is proposed by dividing the global solving domain into several subdomains. The convergence of the loss function is proven. Numerical results are presented to demonstrate the superior accuracy and faster convergence of DANN compared to the traditional physics-informed neural network (PINN).

翻译：本文提出了一种名为双激活神经网络（DANN）的新型网络架构，用于求解含时滞的抛物方程。在DANN中，每个神经元配备两个激活函数以增强网络的非线性表达能力。此外，在两个激活函数之一中引入了一个新参数用于构造二次项，进一步提升了网络捕捉复杂非线性关系的能力。针对解的导数不连续性导致的拟合精度低问题，提出了一种分段拟合方法，将全局求解域划分为若干子域。证明了损失函数的收敛性。数值结果表明，与传统物理信息神经网络（PINN）相比，DANN具有更高的精度和更快的收敛速度。