Data-driven Discovery of Delay Differential Equations with Discrete Delays

TL;DR

This paper tackles data driven discovery of delay differential equations with unknown discrete delays. It combines sparse identification of nonlinear dynamics with an augmented library that includes delayed samples and Bayesian optimization to efficiently infer delays and nonmultiplicative parameters. The approach scales to multiple delays and nonlinear terms such as Hill functions, substantially reducing the number of expensive SINDy evaluations and expanding the set of discoverable models. Numerical experiments on delay logistic, SIR with delay, Mackey-Glass, and multi delay systems demonstrate accurate model recovery and computational advantages, underscoring practical impact for data driven DDE discovery.

Abstract

The Sparse Identification of Nonlinear Dynamics (SINDy) framework is a robust method for identifying governing equations, successfully applied to ordinary, partial, and stochastic differential equations. In this work we extend SINDy to identify delay differential equations by using an augmented library that includes delayed samples and Bayesian optimization. To identify a possibly unknown delay we minimize the reconstruction error over a set of candidates. The resulting methodology improves the overall performance by remarkably reducing the number of calls to SINDy with respect to a brute force approach. We also address a multivariate setting to identify multiple unknown delays and (non-multiplicative) parameters. Several numerical tests on delay differential equations with different long-term behavior, number of variables, delays, and parameters support the use of Bayesian optimization highlighting both the efficacy of the proposed methodology and its computational advantages. As a consequence, the class of discoverable models is significantly expanded.

Figures (6)

• Figure 1: Abstract representation of the information flow to identify DDEs with SINDy. First we collect the time series generated from an unknown process (yellow box), then we enter in the optimization loop (blue boxes). The optimization starts with a candidate delay $\tau$, which is used to form the functions library for the sparse identification problem. The BO loop minimizes the reconstruction error until convergence. The optimized model correctly identifies the unknown delay and can be used for future state predictions (red box).
• Figure 2: Delay logistic equation representative results. The top row shows the test case with $\rho = 1.8$, while in the bottom row we have $\rho = 3.0$. The left column presents the BO evaluations and the identified delays. In the right column we compare the exact solution with the reconstruction corresponding to the identified delay emphasizing the training and the prediction regime.
• Figure 3: SIR model with delay representative results. On the left the BO evaluations and the identified delay. On the right the comparison between the exact solutions and the reconstructions corresponding to the delay $\tau = 1.0$.
• Figure 4: Mackey-Glass equation results for unknown delay $\tau$. On the left the BO evaluations and the identified delay. On the right the comparison between the exact solution and the reconstruction corresponding to the delay $\tau = 1.0$.
• Figure 5: Mackey-Glass equation representative results for unknown delay and Hill coefficient. On the left the BO evaluations and the identified couple of delay and Hill coefficient. On the right the comparison between the exact solution and the reconstruction corresponding to $\tau = 1.0$ and $\alpha = 10$.

Theorems & Definitions (4)

• Example 1: Delay logistic equation
• Example 2: SIR model with delay
• Example 3: Mackey-Glass equation
• Example 4: Multiple delays