Calibrating multi-dimensional complex ODE from noisy data via deep neural networks
This addresses a difficult calibration problem for researchers in fields like biology and finance, though it appears incremental as it builds on existing nonparametric and neural network techniques.
The authors tackled the problem of calibrating complex multi-dimensional ODE systems from noisy data by proposing a two-stage nonparametric method using deep neural networks, achieving consistent recovery without the curse of dimensionality as validated through simulations and application to Covid-19 infection data across 50 U.S. states.
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.