A Deep Neural Network Framework for Solving Forward and Inverse Problems in Delay Differential Equations
This provides a method for researchers and practitioners in mathematical modeling to handle DDEs more flexibly, though it appears incremental as it adapts existing neural network techniques to a specific domain.
The authors tackled solving forward and inverse problems for delay differential equations (DDEs) by proposing a neural delay differential equations (NDDEs) framework based on deep neural networks, which demonstrated high precision in numerical experiments without grid dependence.
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.