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DDE-Find: Learning Delay Differential Equations from Noisy, Limited Data

Robert Stephany

TL;DR

DDE-Find presents an adjoint-based framework to learn parameters, the time delay, and the initial-condition function of delay differential equations from noisy, limited data. The method reduces gradient computation to a forward DDE solve followed by a backward adjoint solve, enabling efficient gradient-based optimization of $\theta$, $\tau$, and $\phi$ via the loss $\mathcal{L}(x)=\int_{0}^{T}\ell(x(t))\,dt+G(x(T))$. It is validated across multiple DDEs, showing accurate recovery of delays and parameters under noise, while revealing identifiability limits for certain initial-condition components and model structures. The work extends NODE-like ideas to DDEs, providing a practical tool for scientific discovery from real-world data with delays, and emphasizes the importance of model structure in parameter identifiability. The open-source implementation enables researchers to apply DDE-Find to diverse delayed systems and explore learning dynamics with noisy measurements.

Abstract

Delay Differential Equations (DDEs) are a class of differential equations that can model diverse scientific phenomena. However, identifying the parameters, especially the time delay, that make a DDE's predictions match experimental results can be challenging. We introduce DDE-Find, a data-driven framework for learning a DDE's parameters, time delay, and initial condition function. DDE-Find uses an adjoint-based approach to efficiently compute the gradient of a loss function with respect to the model parameters. We motivate and rigorously prove an expression for the gradients of the loss using the adjoint. DDE-Find builds upon recent developments in learning DDEs from data and delivers the first complete framework for learning DDEs from data. Through a series of numerical experiments, we demonstrate that DDE-Find can learn DDEs from noisy, limited data.

DDE-Find: Learning Delay Differential Equations from Noisy, Limited Data

TL;DR

DDE-Find presents an adjoint-based framework to learn parameters, the time delay, and the initial-condition function of delay differential equations from noisy, limited data. The method reduces gradient computation to a forward DDE solve followed by a backward adjoint solve, enabling efficient gradient-based optimization of , , and via the loss . It is validated across multiple DDEs, showing accurate recovery of delays and parameters under noise, while revealing identifiability limits for certain initial-condition components and model structures. The work extends NODE-like ideas to DDEs, providing a practical tool for scientific discovery from real-world data with delays, and emphasizes the importance of model structure in parameter identifiability. The open-source implementation enables researchers to apply DDE-Find to diverse delayed systems and explore learning dynamics with noisy measurements.

Abstract

Delay Differential Equations (DDEs) are a class of differential equations that can model diverse scientific phenomena. However, identifying the parameters, especially the time delay, that make a DDE's predictions match experimental results can be challenging. We introduce DDE-Find, a data-driven framework for learning a DDE's parameters, time delay, and initial condition function. DDE-Find uses an adjoint-based approach to efficiently compute the gradient of a loss function with respect to the model parameters. We motivate and rigorously prove an expression for the gradients of the loss using the adjoint. DDE-Find builds upon recent developments in learning DDEs from data and delivers the first complete framework for learning DDEs from data. Through a series of numerical experiments, we demonstrate that DDE-Find can learn DDEs from noisy, limited data.
Paper Structure (21 sections, 2 theorems, 66 equations, 8 figures, 19 tables, 1 algorithm)

This paper contains 21 sections, 2 theorems, 66 equations, 8 figures, 19 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. The Adjoint Equation
  4. Loss
  5. The Adjoint Equation
  6. Methodology
  7. Related Work
  8. Experiments
  9. Delay Exponential Decay Model
  10. Logistic Delay Model
  11. El Niño / Southern Oscillation Model
  12. Cheyne–Stokes Respiration Model
  13. HIV Model
  14. Discussion
  15. Do we Really Need an Adjoint?
  16. ...and 6 more sections

Key Result

Theorem 1

Let $T, \tau > 0$, $\theta \in \mathbb{R}^{p}$, and $\phi \in \mathbb{R}^{q}$. Let $x : [-\tau, T] \to \mathbb{R}^d$ solve the initial value problem in equation eq:IVP. Further, suppose that assumptions assumption:FGell, assumption:X0, and assumption:x hold. Let $\lambda: [0, T] \rightarrow \mathbb{ Then, the derivatives of the loss function, $\mathcal{L}$, with respect to $\theta, \tau$ and $\phi

Figures (8)

  • Figure 1: True, Target, and Predicted trajectories from one of the Delay Exponential Decay experiments with a noise level of $0.3$.
  • Figure 2: True, Target, and Predicted trajectories from one of the Logistic Delay experiments with a noise level of $0.3$.
  • Figure 3: True, Target, and Predicted trajectories from one of the ENSO experiments with a noise level of $0.3$.
  • Figure 4: True, Target, and Predicted trajectories from one of the Cheyne experiments with a noise level of $0.3$.
  • Figure 5: The $T^*$ component of the true, target, and predicted trajectory for the HIV model.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 1