# Interpretable Polynomial Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2208.05072 · HTML: https://mlsanity.com/paper/2208.05072/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 57 references that are in the map.
- First posted on arXiv 9 Aug 2022.

## Builds on (most cited first, 11 of 11)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [On the Power of Over-parametrization in Neural Networks with Quadratic Activation](https://mlsanity.com/paper/1803.01206/) (2018)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Deep Polynomial Neural Networks](https://mlsanity.com/paper/2006.13026/) (2020)
- [Bayesian Neural Ordinary Differential Equations](https://mlsanity.com/paper/2012.07244/) (2020)
- [Universal Approximation with Quadratic Deep Networks](https://mlsanity.com/paper/1808.00098/) (2018)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)

## Built on by (most cited first, 24 of 24)

- [ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEs](https://mlsanity.com/paper/2404.10024/) (2024)
- [Multilinear Operator Networks](https://mlsanity.com/paper/2401.17992/) (2024)
- [Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methods](https://mlsanity.com/paper/2410.05592/) (2024)
- [AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regression](https://mlsanity.com/paper/2312.14237/) (2023)
- [Bayesian polynomial neural networks and polynomial neural ordinary differential equations](https://mlsanity.com/paper/2308.10892/) (2023)
- [Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods](https://mlsanity.com/paper/2412.01181/) (2024)
- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations](https://mlsanity.com/paper/2507.19036/) (2025)
- [Control of Medical Digital Twins with Artificial Neural Networks](https://mlsanity.com/paper/2403.13851/) (2024)
- [A Training Framework for Optimal and Stable Training of Polynomial Neural Networks](https://mlsanity.com/paper/2505.11589/) (2025)
- [The Vanishing Gradient Problem for Stiff Neural Differential Equations](https://mlsanity.com/paper/2508.01519/) (2025)
- [Symbolic Regression via Neural Networks](https://mlsanity.com/paper/2605.04337/) (2026)
- [Interpretable Neural PDE Solvers using Symbolic Frameworks](https://mlsanity.com/paper/2310.20463/) (2023)
- [Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation](https://mlsanity.com/paper/2406.02336/) (2024)
- [Expressive Symbolic Regression for Interpretable Models of Discrete-Time Dynamical Systems](https://mlsanity.com/paper/2406.06585/) (2024)
- [Learning dynamical systems with biochemically informed neural ordinary differential equations](https://mlsanity.com/paper/2605.24170/) (2026)
- [Neural Astrophysical Wind Models](https://mlsanity.com/paper/2306.11666/) (2023)
- [TMPNN: High-Order Polynomial Regression Based on Taylor Map Factorization](https://mlsanity.com/paper/2307.16105/) (2023)
- [Statistical Mechanics and Artificial Neural Networks: Principles, Models, and Applications](https://mlsanity.com/paper/2405.10957/) (2024)
- [Control of dynamical systems with neural networks](https://mlsanity.com/paper/2510.12810/) (2025)
- [CODE: A global approach to ODE dynamics learning](https://mlsanity.com/paper/2511.15619/) (2025)
- [Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error](https://mlsanity.com/paper/2605.31438/) (2026)
- [Symbolic Neural ODEs: Learning interpretable models from time-series data](https://mlsanity.com/paper/2608.22112/) (2026)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

Neural networks have the ability to serve as universal function approximators, but they are not interpretable and don't generalize well outside of their training region. Both of these issues are problematic when trying to apply standard neural ordinary differential equations (neural ODEs) to dynamical systems. We introduce the polynomial neural ODE, which is a deep polynomial neural network inside of the neural ODE framework. We demonstrate the capability of polynomial neural ODEs to predict outside of the training region, as well as perform direct symbolic regression without additional tools such as SINDy.