Double-activation neural network for solving parabolic equations with time delay

TL;DR

This paper introduces the double-activation neural network (DANN), a new architecture for solving parabolic equations with time delay, featuring enhanced nonlinear capacity and a piecewise fitting approach for improved accuracy.

Contribution

The paper proposes DANN with dual activation functions and a novel quadratic parameter, along with a piecewise fitting method, advancing neural network solutions for delayed parabolic equations.

Findings

DANN achieves higher accuracy than traditional PINN.

DANN converges faster in numerical experiments.

Piecewise fitting improves solution quality.

Abstract

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).