Ordinary differential equations (ODEs) are widely used to model complex dynamics that arises in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally very difficult. In this work, we propose a two-stage nonparametric approach to address this problem. We first extract the de-noised data and their higher order derivatives using boundary kernel method, and then feed them into a sparsely connected deep neural network with ReLU activation function. Our method is able to recover the ODE system without being subject to the curse of dimensionality and complicated ODE structure. When the ODE possesses a general modular structure, with each modular component involving only a few input variables, and the network architecture is properly chosen, our method is proven to be consistent. Theoretical properties are corroborated by an extensive simulation study that demonstrates the validity and effectiveness of the proposed method. Finally, we use our method to simultaneously characterize the growth rate of Covid-19 infection cases from 50 states of the USA.

翻译：常微分方程广泛应用于模拟生物学、化学、工程、金融、物理学等领域中出现的复杂动力学过程。利用含噪数据标定复杂常微分方程系统通常极为困难。本文提出一种两阶段非参数方法来解决该问题。首先，我们采用边界核方法提取去噪数据及其高阶导数，然后将去噪数据输入采用ReLU激活函数的稀疏连接深度神经网络。该方法能够在不受维数诅咒和复杂常微分方程结构制约的前提下恢复微分方程系统。当常微分方程具有一般模块化结构（每个模块仅涉及少数输入变量）且网络结构选择适当时，本文证明了该方法具有一致性。理论性质通过一项广泛模拟研究得到验证，该研究展示了所提方法的有效性与准确性。最后，我们应用该方法同时刻画了美国50个州新冠肺炎感染病例的增长率特征。