# Learning Fine Scale Dynamics from Coarse Observations via Inner Recurrence

arXiv: https://arxiv.org/abs/2206.01807 · HTML: https://mlsanity.com/paper/2206.01807/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 39 references that are in the map.
- First posted on arXiv 3 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (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)
- [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)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Learning reduced systems via deep neural networks with memory](https://mlsanity.com/paper/2003.09451/) (2020)
- [On generalized residue network for deep learning of unknown dynamical systems](https://mlsanity.com/paper/2002.02528/) (2020)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)
- [Data-driven learning of non-autonomous systems](https://mlsanity.com/paper/2006.02392/) (2020)
- [Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learning](https://mlsanity.com/paper/2203.03458/) (2022)

## Built on by (most cited first, 1 of 1)

- [Deep learning for model correction of dynamical systems with data scarcity](https://mlsanity.com/paper/2410.17913/) (2024)

## Abstract (arXiv)

Recent work has focused on data-driven learning of the evolution of unknown systems via deep neural networks (DNNs), with the goal of conducting long term prediction of the dynamics of the unknown system. In many real-world applications, data from time-dependent systems are often collected on a time scale that is coarser than desired, due to various restrictions during the data acquisition process. Consequently, the observed dynamics can be severely under-sampled and do not reflect the true dynamics of the underlying system. This paper presents a computational technique to learn the fine-scale dynamics from such coarsely observed data. The method employs inner recurrence of a DNN to recover the fine-scale evolution operator of the underlying system. In addition to mathematical justification, several challenging numerical examples, including unknown systems of both ordinary and partial differential equations, are presented to demonstrate the effectiveness of the proposed method.