Developing a Physics-Informed Neural Network to Estimate the Parameters of a Delay Differential Equation

Abstract Delay differential equations (DDEs) arise naturally in physics, engineering, and the life sciences, for example, in lasers with delayed optical feedback, machining chatter, and population dynamics. Unlike ordinary differential equations, they require a history function and are more challenging to solve numerically. Physics-informed neural networks (PINNs) incorporate governing equations into the training loss of a neural network, making them well suited to problems that combine measured data with unknown parameters. We use the delayed logistic equation, fitted to Nicholson's blowfly data, as a test case for applying a PINN to a DDE. The model is implemented in MATLAB with the Deep Learning Toolbox [1, 2]. The network learns the delay (τ), growth rate (r), and carrying capacity (K), together with the population curve. We compare the results obtained using different physics weights with a solution from the traditional solver dde23. The learned parameters (τ ≈ 8.8 days, rτ ≈ 1.8, K ≈ 2400) are close to published values, and the two solutions agree in terms of period and amplitude. We also observe that the PINN is reliable only within the time range over which the governing equation was enforced.