mlsanityData to 14 Oct 2026

Bayesian polynomial neural networks and polynomial neural ordinary differential equations

arXiv 2308.10892 ↗First posted 17 Aug 2023cs.LG cs.AI cs.CEcited by 6

Not on the map. Why?

Abstract (arXiv)

Symbolic regression with polynomial neural networks and polynomial neural ordinary differential equations (ODEs) are two recent and powerful approaches for equation recovery of many science and engineering problems. However, these methods provide point estimates for the model parameters and are currently unable to accommodate noisy data. We address this challenge by developing and validating the following Bayesian inference methods: the Laplace approximation, Markov Chain Monte Carlo (MCMC) sampling methods, and variational inference. We have found the Laplace approximation to be the best method for this class of problems. Our work can be easily extended to the broader class of symbolic neural networks to which the polynomial neural network belongs.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 26 of its 126 references that are in the map. First posted on arXiv 17 Aug 2023.

Builds on

26 of 126 references.

References · 26

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2018Neural Ordinary Differential Equationscited by 3,779
  3. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  4. 2020Neural Controlled Differential Equations for Irregular Time Seriescited by 506
  5. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  6. 2020Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Userscited by 338
  7. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  8. 2022On Neural Differential Equationscited by 268
  9. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  10. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 161
  11. 2019Neural Jump Stochastic Differential Equationscited by 150
  12. 2020Neural Rough Differential Equations for Long Time Seriescited by 143
14 more
  1. 2020SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamicscited by 125
  2. 2019Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discoverycited by 93
  3. 2021Stiff Neural Ordinary Differential Equationscited by 78
  4. 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
  5. 2020Deep Polynomial Neural Networkscited by 63
  6. 2020Bayesian Neural Ordinary Differential Equationscited by 45
  7. 2018Universal Approximation with Quadratic Deep Networkscited by 29
  8. 2020Autonomous Discovery of Unknown Reaction Pathways from Data by Chemical Reaction Neural Networkcited by 27
  9. 2022Interpretable Polynomial Neural Ordinary Differential Equationscited by 24
  10. 2022Deep Learning and Symbolic Regression for Discovering Parametric Equationscited by 23
  11. 2021Symbolic regression for scientific discovery: an application to wind speed forecastingcited by 16
  12. 2017Internet of Things: An Overviewcited by 14
  13. 2023Toward Physically Plausible Data-Driven Models: A Novel Neural Network Approach to Symbolic Regressioncited by 14
  14. 2023Uncertainty and Structure in Neural Ordinary Differential Equationscited by 6

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 2; 2025: 2; 2026: 2.

6 papers.

  1. 2024Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methodscited by 11
  2. 2024Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methodscited by 5
  3. 2025The Vanishing Gradient Problem for Stiff Neural Differential Equationscited by 3
  4. 2025Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamicscited by 1
  5. 2026Deep Variational Inference Symbolic Regressioncited by 0
  6. 2026Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysiscited by 0