mlsanityData to 7 Oct 2026

Interpretable Polynomial Neural Ordinary Differential Equations

arXiv 2208.05072 ↗First posted 9 Aug 2022cs.LG cs.AIcited by 24

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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.

Key facts. Cited by 24 papers in the map (data through 7 Oct 2026). Builds on 11 of its 57 references that are in the map. First posted on arXiv 9 Aug 2022.

Builds on

11 of 57 references.

References · 11

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  3. 2020Scalable Gradients for Stochastic Differential Equationscited by 264
  4. 2018On the Power of Over-parametrization in Neural Networks with Quadratic Activationcited by 102
  5. 2021PySINDy: A comprehensive Python package for robust sparse system identificationcited by 88
  6. 2016Why Deep Neural Networks for Function Approximation?cited by 82
  7. 2021Stiff Neural Ordinary Differential Equationscited by 77
  8. 2020Deep Polynomial Neural Networkscited by 63
  9. 2020Bayesian Neural Ordinary Differential Equationscited by 44
  10. 2018Universal Approximation with Quadratic Deep Networkscited by 29
  11. 2021Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modelingcited by 27

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 5; 2024: 8; 2025: 6; 2026: 5.

24 papers.

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  1. 2024ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEscited by 78
  2. 2024Multilinear Operator Networkscited by 12
  3. 2024Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methodscited by 11
  4. 2023AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regressioncited by 9
  5. 2023Bayesian polynomial neural networks and polynomial neural ordinary differential equationscited by 6
  6. 2024Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methodscited by 5
  7. 2025Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamicscited by 4
  8. 2025Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcationscited by 4
  9. 2024Control of Medical Digital Twins with Artificial Neural Networkscited by 3
  10. 2025The Vanishing Gradient Problem for Stiff Neural Differential Equationscited by 3
  11. 2025A Training Framework for Optimal and Stable Training of Polynomial Neural Networkscited by 2
  12. 2026Symbolic Regression via Neural Networkscited by 2
12 more
  1. 2023Interpretable Neural PDE Solvers using Symbolic Frameworkscited by 1
  2. 2024Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximationcited by 1
  3. 2024Expressive Symbolic Regression for Interpretable Models of Discrete-Time Dynamical Systemscited by 1
  4. 2023Neural Astrophysical Wind Modelscited by 0
  5. 2023TMPNN: High-Order Polynomial Regression Based on Taylor Map Factorizationcited by 0
  6. 2024Statistical Mechanics and Artificial Neural Networks: Principles, Models, and Applicationscited by 0
  7. 2025Control of dynamical systems with neural networkscited by 0
  8. 2025CODE: A global approach to ODE dynamics learningcited by 0
  9. 2026Learning dynamical systems with biochemically informed neural ordinary differential equationscited by 0
  10. 2026Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training errorcited by 0
  11. 2026Symbolic Neural ODEs: Learning interpretable models from time-series datacited by 0
  12. 2026Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysiscited by 0