# Modular Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2109.07359 · HTML: https://mlsanity.com/paper/2109.07359/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 12 references that are in the map.
- First posted on arXiv 15 Sep 2021.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [On Second Order Behaviour in Augmented Neural ODEs](https://mlsanity.com/paper/2006.07220/) (2020)

## Built on by (most cited first, 2 of 2)

- [Enhancing Trust in LLMs: Algorithms for Comparing and Interpreting LLMs](https://mlsanity.com/paper/2406.01943/) (2024)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)

## Abstract (arXiv)

The laws of physics have been written in the language of dif-ferential equations for centuries. Neural Ordinary Differen-tial Equations (NODEs) are a new machine learning architecture which allows these differential equations to be learned from a dataset. These have been applied to classical dynamics simulations in the form of Lagrangian Neural Net-works (LNNs) and Second Order Neural Differential Equations (SONODEs). However, they either cannot represent the most general equations of motion or lack interpretability. In this paper, we propose Modular Neural ODEs, where each force component is learned with separate modules. We show how physical priors can be easily incorporated into these models. Through a number of experiments, we demonstrate these result in better performance, are more interpretable, and add flexibility due to their modularity.