Calibrating multi-dimensional complex ODE from noisy data via deep neural networks

Kexuan Li, Fangfang Wang, Ruiqi Liu, Fan Yang, Zuofeng Shang

Research output: Contribution to journalArticlepeer-review


Ordinary differential equations (ODEs) are widely used to model complex dynamics that arise in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally challenging. In this paper, 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 rectified linear unit (ReLU) activation function. Our method is able to recover the ODE system without being subject to the curse of dimensionality and the complexity of the ODE structure. We have shown that our method is consistent if the ODE possesses a general modular structure with each modular component involving only a few input variables, and the network architecture is properly chosen. Theoretical properties are corroborated by an extensive simulation study that also demonstrates the effectiveness of the proposed method in finite samples. Finally, we use our method to simultaneously characterize the growth rate of COVID-19 cases from the 50 states of the United States.

Original languageEnglish (US)
Article number106147
JournalJournal of Statistical Planning and Inference
StatePublished - Sep 2024

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


  • Deep neural networks
  • Feature selection
  • Ordinary differential equations
  • ReLU activation function


Dive into the research topics of 'Calibrating multi-dimensional complex ODE from noisy data via deep neural networks'. Together they form a unique fingerprint.

Cite this