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Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying…

Numerical Analysis · Mathematics 2025-06-18 Matteo Caldana , Jan S. Hesthaven

Neural ordinary differential equations (ODEs) have attracted much attention as continuous-time counterparts of deep residual neural networks (NNs), and numerous extensions for recurrent NNs have been proposed. Since the 1980s, ODEs have…

Machine Learning · Computer Science 2022-10-17 Kazuki Irie , Francesco Faccio , Jürgen Schmidhuber

Neural ordinary differential equations (Neural ODEs) are an effective framework for learning dynamical systems from irregularly sampled time series data. These models provide a continuous-time latent representation of the underlying…

Machine Learning · Computer Science 2023-03-06 Edward De Brouwer , Rahul G. Krishnan

Neural Ordinary Differential Equations (Neural ODEs) represent a significant breakthrough in deep learning, promising to bridge the gap between machine learning and the rich theoretical frameworks developed in various mathematical fields…

Machine Learning · Computer Science 2024-09-24 Jaouad Dabounou

Neural differential equations are a promising new member in the neural network family. They show the potential of differential equations for time series data analysis. In this paper, the strength of the ordinary differential equation (ODE)…

Machine Learning · Computer Science 2020-05-21 Mansura Habiba , Barak A. Pearlmutter

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…

Machine Learning · Computer Science 2023-08-21 Colby Fronk , Linda Petzold

Neural Ordinary Differential Equations (N-ODEs) are a powerful building block for learning systems, which extend residual networks to a continuous-time dynamical system. We propose a Bayesian version of N-ODEs that enables well-calibrated…

Machine Learning · Computer Science 2020-02-19 Andreas Look , Melih Kandemir

Neural ordinary differential equations (ODEs) have been attracting increasing attention in various research domains recently. There have been some works studying optimization issues and approximation capabilities of neural ODEs, but their…

Machine Learning · Computer Science 2022-03-04 Hanshu Yan , Jiawei Du , Vincent Y. F. Tan , Jiashi Feng

This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and…

Machine Learning · Computer Science 2024-01-09 Lars Ruthotto

The existing Neural ODE formulation relies on an explicit knowledge of the termination time. We extend Neural ODEs to implicitly defined termination criteria modeled by neural event functions, which can be chained together and…

Machine Learning · Computer Science 2021-10-28 Ricky T. Q. Chen , Brandon Amos , Maximilian Nickel

Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…

Machine Learning · Computer Science 2025-03-27 C. Coelho , M. Fernanda P. Costa , L. L. Ferrás

Recently, Neural Ordinary Differential Equations has emerged as a powerful framework for modeling physical simulations without explicitly defining the ODEs governing the system, but instead learning them via machine learning. However, the…

Differential equations are frequently used in engineering domains, such as modeling and control of industrial systems, where safety and performance guarantees are of paramount importance. Traditional physics-based modeling approaches…

Systems and Control · Electrical Eng. & Systems 2020-11-30 Aaron Tuor , Jan Drgona , Draguna Vrabie

Neural Ordinary Differential Equations (Neural ODEs) are the continuous analog of Residual Neural Networks (ResNets). We investigate whether the discrete dynamics defined by a ResNet are close to the continuous one of a Neural ODE. We first…

Machine Learning · Computer Science 2022-09-16 Michael E. Sander , Pierre Ablin , Gabriel Peyré

Neural ordinary differential equations describe how values change in time. This is the reason why they gained importance in modeling sequential data, especially when the observations are made at irregular intervals. In this paper we propose…

Machine Learning · Computer Science 2021-10-26 Marin Biloš , Johanna Sommer , Syama Sundar Rangapuram , Tim Januschowski , Stephan Günnemann

Neural Ordinary Differential Equations (NODEs) use a neural network to model the instantaneous rate of change in the state of a system. However, despite their apparent suitability for dynamics-governed time-series, NODEs present a few…

Machine Learning · Computer Science 2021-08-18 Alexander Norcliffe , Cristian Bodnar , Ben Day , Jacob Moss , Pietro Liò

We show that Neural Ordinary Differential Equations (ODEs) learn representations that preserve the topology of the input space and prove that this implies the existence of functions Neural ODEs cannot represent. To address these…

Machine Learning · Statistics 2019-10-29 Emilien Dupont , Arnaud Doucet , Yee Whye Teh

Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had…

Machine Learning · Computer Science 2024-05-07 Christina Runkel , Ander Biguri , Carola-Bibiane Schönlieb

Neural ODEs (NODEs) have emerged as powerful tools for modeling time series data, offering the flexibility to adapt to varying input scales and capture complex dynamics. However, they face significant challenges: first, their reliance on…

Machine Learning · Computer Science 2025-10-07 Muhao Guo , Yang Weng

Neural Ordinary Differential Equations (NODEs) have proven to be a powerful modeling tool for approximating (interpolation) and forecasting (extrapolation) irregularly sampled time series data. However, their performance degrades…

Machine Learning · Computer Science 2020-04-29 Hammad A. Ayyubi , Yi Yao , Ajay Divakaran
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