A Deep Neural Network Framework for Solving Forward and Inverse Problems in Delay Differential Equations

TL;DR

This paper introduces NDDEs, a neural network-based framework that effectively solves forward and inverse delay differential equations without traditional grid methods, demonstrating high accuracy and potential in modeling delays.

Contribution

The paper presents a novel neural network framework for delay differential equations that handles both forward and inverse problems, including parameter estimation from data.

Findings

High precision in solving forward delay differential equations.

Effective inverse problem solving with delay parameter estimation.

Demonstrated potential for practical mathematical modeling.

Abstract

We propose a unified framework for delay differential equations (DDEs) based on deep neural networks (DNNs) - the neural delay differential equations (NDDEs), aimed at solving the forward and inverse problems of delay differential equations. This framework could embed delay differential equations into neural networks to accommodate the diverse requirements of DDEs in terms of initial conditions, control equations, and known data. NDDEs adjust the network parameters through automatic differentiation and optimization algorithms to minimize the loss function, thereby obtaining numerical solutions to the delay differential equations without the grid dependence and polynomial interpolation typical of traditional numerical methods. In addressing inverse problems, the NDDE framework can utilize observational data to perform precise estimation of single or multiple delay parameters, which is very important in practical mathematical modeling. The results of multiple numerical experiments have shown that NDDEs demonstrate high precision in both forward and inverse problems, proving their effectiveness and promising potential in dealing with delayed differential equation issues.