# Learning Subgrid-scale Models with Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2212.09967 · HTML: https://mlsanity.com/paper/2212.09967/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 51 references that are in the map.
- First posted on arXiv 20 Dec 2022.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [MALI: A memory efficient and reverse accurate integrator for Neural ODEs](https://mlsanity.com/paper/2102.04668/) (2021)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [Will Artificial Intelligence supersede Earth System and Climate Models?](https://mlsanity.com/paper/2101.09126/) (2021)
- [Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs](https://mlsanity.com/paper/2201.05715/) (2022)
- [Neural ODEs as the Deep Limit of ResNets with constant weights](https://mlsanity.com/paper/1906.12183/) (2019)
- [Improving Robustness and Uncertainty Modelling in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2112.12707/) (2021)
- [Differentiable physics-enabled closure modeling for Burgers' turbulence](https://mlsanity.com/paper/2209.11614/) (2022)
- [InfoCNF: An Efficient Conditional Continuous Normalizing Flow with Adaptive Solvers](https://mlsanity.com/paper/1912.03978/) (2019)

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

- [NORi: An ML-Augmented Ocean Boundary Layer Parameterization](https://mlsanity.com/paper/2512.04452/) (2025)
- [Differentiable DG with Neural Operator Source Term Correction](https://mlsanity.com/paper/2310.18897/) (2023)
- [A Priori Uncertainty Quantification of Reacting Turbulence Closure Models using Bayesian Neural Networks](https://mlsanity.com/paper/2402.18729/) (2024)
- [Machine-Learned Closure of URANS for Stably Stratified Turbulence: Connecting Physical Timescales & Data Hyperparameters of Deep Time-Series Models](https://mlsanity.com/paper/2404.16141/) (2024)

## Abstract (arXiv)

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection-diffusion PDE, and the viscous Burgers' PDE are used to illustrate this approach.