Dynamic systems described by differential equations often involve feedback among system components. When there are time delays for components to sense and respond to feedback, delay differential equation (DDE) models are commonly used. This paper considers the problem of inferring unknown system parameters, including the time delays, from noisy and sparse experimental data observed from the system. We propose an extension of manifold-constrained Gaussian processes to conduct parameter inference for DDEs, whereas the time delay parameters have posed a challenge for existing methods that bypass numerical solvers. Our method uses a Bayesian framework to impose a Gaussian process model over the system trajectory, conditioned on the manifold constraint that satisfies the DDEs. For efficient computation, a linear interpolation scheme is developed to approximate the values of the time-delayed system outputs, along with corresponding theoretical error bounds on the approximated derivatives. Two simulation examples, based on Hutchinson's equation and the lac operon system, together with a real-world application using Ontario COVID-19 data, are used to illustrate the efficacy of our method.

翻译：由微分方程描述的动态系统通常涉及系统组件之间的反馈。当组件感知和响应反馈存在时间延迟时，通常采用延迟微分方程模型。本文考虑从系统观测到的噪声稀疏实验数据中推断未知系统参数（包括时间延迟）的问题。我们提出了流形约束高斯过程的一种扩展方法，用于对延迟微分方程进行参数推断，而时间延迟参数对现有绕过数值求解器的方法构成了挑战。我们的方法采用贝叶斯框架，在满足延迟微分方程的流形约束条件下，对系统轨迹施加高斯过程模型。为实现高效计算，我们开发了一种线性插值方案来近似时滞系统输出的值，并给出了近似导数的相应理论误差界。基于哈钦森方程和乳糖操纵子系统的两个仿真示例，以及使用安大略省新冠疫情数据的实际应用，共同验证了我们方法的有效性。