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

TL;DR

The paper tackles solving forward and inverse problems for delay differential equations (DDEs) by proposing Neural Delay Differential Equations (NDDEs), a framework that embeds DDEs into deep neural networks and uses automatic differentiation to compute derivatives. It defines specialized loss functions for residual dynamics, initial conditions, and data-fitting, with adaptive weighting to balance training across terms, and extends to systems of DDEs via a multi-network approach. Empirical results on single-delay and multi-delay DDEs demonstrate high accuracy in both forward solution and parameter/delay estimation, with relative errors often well below 1-3%. The work provides a discretization-free, continuous-differentiable alternative to classical methods and includes public Python/JAX implementations to facilitate adoption in practice.

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.

Figures (21)

• Figure 1: The framework of NDDEs for solving forward problem of delay differential equations.
• Figure 2: The framework of NDDEs for solving inverse problem of delay differential equations.
• Figure 3: The framework of NDDEs for solving forward problem of the system of delay differential equations.
• Figure 4: The framework of NDDEs for solving inverse problem of the system of delay differential equations.
• Figure 5: Trajectories of the final predicted vs. true trajectories for the delay differential equation with $\tau=0.5$ as shown in Eq. \ref{['equ17a']}.