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

arXiv: https://arxiv.org/abs/2106.03591 · HTML: https://mlsanity.com/paper/2106.03591/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 53 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [LassoNet: A Neural Network with Feature Sparsity](https://mlsanity.com/paper/1907.12207/) (2019)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [Deep Neural Networks for Estimation and Inference](https://mlsanity.com/paper/1809.09953/) (2018)
- [Deep Learning in Bioinformatics](https://mlsanity.com/paper/1603.06430/) (2016)
- [Facial Expression Analysis under Partial Occlusion: A Survey](https://mlsanity.com/paper/1802.08784/) (2018)
- [On the Curse of Memory in Recurrent Neural Networks: Approximation and Optimization Analysis](https://mlsanity.com/paper/2009.07799/) (2020)
- [On the rate of convergence of fully connected very deep neural network regression estimates](https://mlsanity.com/paper/1908.11133/) (2019)
- [On Deep Instrumental Variables Estimate](https://mlsanity.com/paper/2004.14954/) (2020)
- [Network Reconstruction From High Dimensional Ordinary Differential Equations](https://mlsanity.com/paper/1610.03177/) (2016)
- [Optimal Nonparametric Inference via Deep Neural Network](https://mlsanity.com/paper/1902.01687/) (2019)
- [Estimation of the Mean Function of Functional Data via Deep Neural Networks](https://mlsanity.com/paper/2012.04573/) (2020)

## Built on by (most cited first, 6 of 6)

- [Deep Feature Screening: Feature Selection for Ultra High-Dimensional Data via Deep Neural Networks](https://mlsanity.com/paper/2204.01682/) (2022)
- [Deep Neural Network Classifier for Multi-dimensional Functional Data](https://mlsanity.com/paper/2205.08592/) (2022)
- [Variable selection for nonlinear Cox regression model via deep learning](https://mlsanity.com/paper/2211.09287/) (2022)
- [Semiparametric Regression for Spatial Data via Deep Learning](https://mlsanity.com/paper/2301.03747/) (2023)
- [Nonparametric regression with modified ReLU networks](https://mlsanity.com/paper/2207.08306/) (2022)
- [Deep Learning for Efficient GWAS Feature Selection](https://mlsanity.com/paper/2312.15055/) (2023)

## Abstract (arXiv)

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.