# Neural Ordinary Differential Equations for Data-Driven Reduced Order Modeling of Environmental Hydrodynamics

arXiv: https://arxiv.org/abs/2104.13962 · HTML: https://mlsanity.com/paper/2104.13962/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 80 references that are in the map.
- First posted on arXiv 22 Apr 2021.

## Builds on (most cited first, 14 of 14)

- [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)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data](https://mlsanity.com/paper/1908.03190/) (2019)
- [Learning functionals via LSTM neural networks for predicting vessel dynamics in extreme sea states](https://mlsanity.com/paper/1912.13382/) (2019)
- [A greedy non-intrusive reduced order model for shallow water equations](https://mlsanity.com/paper/2002.11329/) (2020)

## Built on by (most cited first, 5 of 5)

- [Proximal Implicit ODE Solvers for Accelerating Learning Neural ODEs](https://mlsanity.com/paper/2204.08621/) (2022)
- [Data-driven reduced order modeling of environmental hydrodynamics using deep autoencoders and neural ODEs](https://mlsanity.com/paper/2107.02784/) (2021)
- [Learning POD of Complex Dynamics Using Heavy-ball Neural ODEs](https://mlsanity.com/paper/2202.12373/) (2022)
- [Deep Convolutional Architectures for Extrapolative Forecast in Time-dependent Flow Problems](https://mlsanity.com/paper/2209.09651/) (2022)
- [Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry](https://mlsanity.com/paper/2607.00669/) (2026)

## Abstract (arXiv)

Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. Here, we explore the use of Neural Ordinary Differential Equations, a recently introduced family of continuous-depth, differentiable networks (Chen et al 2018), as a way to propagate latent-space dynamics in reduced order models. We compare their behavior with two classical non-intrusive methods based on proper orthogonal decomposition and radial basis function interpolation as well as dynamic mode decomposition. The test problems we consider include incompressible flow around a cylinder as well as real-world applications of shallow water hydrodynamics in riverine and estuarine systems. Our findings indicate that Neural ODEs provide an elegant framework for stable and accurate evolution of latent-space dynamics with a promising potential of extrapolatory predictions. However, in order to facilitate their widespread adoption for large-scale systems, significant effort needs to be directed at accelerating their training times. This will enable a more comprehensive exploration of the hyperparameter space for building generalizable Neural ODE approximations over a wide range of system dynamics.