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.

翻译：延迟微分方程（DDEs）是一类能够刻画多种科学现象的微分方程。然而，确定使DDE预测与实验结果相匹配的参数（尤其是时延）可能极具挑战性。我们提出DDE-Find——一种数据驱动的框架，用于学习DDE的参数、时延和初始条件函数。DDE-Find采用伴随方法，高效计算损失函数关于模型参数的梯度。我们基于伴随方法推导并严格证明了损失函数梯度的表达式。DDE-Find在从数据中学习DDE的最新进展基础上，首次构建了从数据中完整学习DDE的理论框架。通过一系列数值实验，我们证明DDE-Find能够从含噪且有限的数据中有效学习DDE。