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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 (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

By interpreting the forward dynamics of the latent representation of neural networks as an ordinary differential equation, Neural Ordinary Differential Equation (Neural ODE) emerged as an effective framework for modeling a system dynamics…

Machine Learning · Computer Science 2020-10-19 Daehoon Gwak , Gyuhyeon Sim , Michael Poli , Stefano Massaroli , Jaegul Choo , Edward Choi

Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in…

Numerical Analysis · Mathematics 2026-05-11 Andrew Tagg , Andrew Frandsen , Andrew Ning

Neural ordinary differential equations (Neural ODEs) is a class of machine learning models that approximate the time derivative of hidden states using a neural network. They are powerful tools for modeling continuous-time dynamical systems,…

Machine Learning · Statistics 2024-07-16 Wenbo Hao

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

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 Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with some representative datasets. Recently, an augmented framework has been…

Machine Learning · Computer Science 2021-02-23 Qunxi Zhu , Yao Guo , Wei Lin

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

The order/dimension of models derived on the basis of data is commonly restricted by the number of observations, or in the context of monitored systems, sensing nodes. This is particularly true for structural systems (e.g., civil or…

Machine Learning · Computer Science 2022-12-01 Zhilu Lai , Wei Liu , Xudong Jian , Kiran Bacsa , Limin Sun , Eleni Chatzi

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

The neural Ordinary Differential Equation (ODE) model has shown success in learning complex continuous-time processes from observations on discrete time stamps. In this work, we consider the modeling and forecasting of time series data that…

Machine Learning · Statistics 2023-06-05 Yixuan Tan , Liyan Xie , Xiuyuan Cheng

Embedding nonlinear dynamical systems into artificial neural networks is a powerful new formalism for machine learning. By parameterizing ordinary differential equations (ODEs) as neural network layers, these Neural ODEs are…

Machine Learning · Computer Science 2024-10-28 Mikko Lehtimäki , Lassi Paunonen , Marja-Leena Linne

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

The neural ordinary differential equation (ODE) framework has emerged as a powerful tool for developing accelerated surrogate models of complex physical systems governed by partial differential equations (PDEs). A popular approach for PDE…

Fluid Dynamics · Physics 2025-03-26 Ashish S. Nair , Shivam Barwey , Pinaki Pal , Jonathan F. MacArt , Troy Arcomano , Romit Maulik

Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. Here, we explore the use of Neural Ordinary Differential Equations, a recently introduced family of…

Machine Learning · Computer Science 2021-04-30 Sourav Dutta , Peter Rivera-Casillas , Matthew W. Farthing

Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with representative datasets. Recently, an augmented framework has been…

Machine Learning · Computer Science 2023-04-12 Qunxi Zhu , Yao Guo , Wei Lin

Ordinary differential equations (ODEs) can provide mechanistic models of temporally local changes of processes, where parameters are often informed by external knowledge. While ODEs are popular in systems modeling, they are less established…

Methodology · Statistics 2025-07-10 Maren Hackenberg , Astrid Pechmann , Clemens Kreutz , Janbernd Kirschner , Harald Binder

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

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
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