# Modeling unknown dynamical systems with hidden parameters

arXiv: https://arxiv.org/abs/2202.01858 · HTML: https://mlsanity.com/paper/2202.01858/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 26 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data](https://mlsanity.com/paper/1908.03190/) (2019)
- [Numerical Aspects for Approximating Governing Equations Using Data](https://mlsanity.com/paper/1809.09170/) (2018)
- [Learning reduced systems via deep neural networks with memory](https://mlsanity.com/paper/2003.09451/) (2020)
- [Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data](https://mlsanity.com/paper/1905.10396/) (2019)
- [On generalized residue network for deep learning of unknown dynamical systems](https://mlsanity.com/paper/2002.02528/) (2020)
- [Data-driven learning of non-autonomous systems](https://mlsanity.com/paper/2006.02392/) (2020)

## Built on by (most cited first, 4 of 4)

- [Finite Expression Methods for Discovering Physical Laws from Data](https://mlsanity.com/paper/2305.08342/) (2023)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [InVAErt networks: a data-driven framework for model synthesis and identifiability analysis](https://mlsanity.com/paper/2307.12586/) (2023)
- [Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamics](https://mlsanity.com/paper/2510.07549/) (2025)

## Abstract (arXiv)

We present a data-driven numerical approach for modeling unknown dynamical systems with missing/hidden parameters. The method is based on training a deep neural network (DNN) model for the unknown system using its trajectory data. A key feature is that the unknown dynamical system contains system parameters that are completely hidden, in the sense that no information about the parameters is available through either the measurement trajectory data or our prior knowledge of the system. We demonstrate that by training a DNN using the trajectory data with sufficient time history, the resulting DNN model can accurately model the unknown dynamical system. For new initial conditions associated with new, and unknown, system parameters, the DNN model can produce accurate system predictions over longer time.