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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

We introduce Neural Dynamical Systems (NDS), a method of learning dynamical models in various gray-box settings which incorporates prior knowledge in the form of systems of ordinary differential equations. NDS uses neural networks to…

The accurate forecasting of complex, high-dimensional dynamical systems from observational data is a fundamental task across numerous scientific and engineering disciplines. A significant challenge arises from noise-corrupted measurements,…

Machine Learning · Computer Science 2026-04-14 Xuyang Li , John Harlim , Dibyajyoti Chakraborty , Romit Maulik

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

Solution of Ordinary Differential Equation (ODE) model of dynamical system may not agree with its observed values. Often this discrepancy can be attributed to unmodeled forcings in the evolution rule of the dynamical system. In this…

Computational Engineering, Finance, and Science · Computer Science 2021-08-13 Saurabh Dixit , Soumyendu Raha

The last decade witnessed a rise in the importance of supervised learning applications involving {\em big data} and {\em big models}. Big data refers to situations where the amounts of training data available and needed causes difficulties…

Optimization and Control · Mathematics 2018-11-01 Konstantin Mishchenko , Peter Richtárik

Robust 3D object detection remains a pivotal concern in the domain of autonomous field robotics. Despite notable enhancements in detection accuracy across standard datasets, real-world urban environments, characterized by their unstructured…

Robotics · Computer Science 2024-05-14 Houze Liu , Chongqing Wang , Xiaoan Zhan , Haotian Zheng , Chang Che

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

Starting from a heuristic learning scheme for N-person games, we derive a new class of continuous-time learning dynamics consisting of a replicator-like drift adjusted by a penalty term that renders the boundary of the game's strategy space…

Optimization and Control · Mathematics 2014-04-08 Pierre Coucheney , Bruno Gaujal , Panayotis Mertikopoulos

End-to-end learning of dynamical systems with black-box models, such as neural ordinary differential equations (ODEs), provides a flexible framework for learning dynamics from data without prescribing a mathematical model for the dynamics.…

Machine Learning · Statistics 2022-06-20 Paidamoyo Chapfuwa , Sherri Rose , Lawrence Carin , Edward Meeds , Ricardo Henao

Reinforcement learning (RL) is attracting attention as an effective way to solve sequential optimization problems that involve high dimensional state/action space and stochastic uncertainties. Many such problems involve constraints…

Machine Learning · Computer Science 2021-04-01 Haeun Yoo , Victor M. Zavala , Jay H. Lee

Humans and animals can learn complex predictive models that allow them to accurately and reliably reason about real-world phenomena, and they can adapt such models extremely quickly in the face of unexpected changes. Deep neural network…

Machine Learning · Computer Science 2019-01-30 Anusha Nagabandi , Chelsea Finn , Sergey Levine

In recent years, deep learning has been connected with optimal control as a way to define a notion of a continuous underlying learning problem. In this view, neural networks can be interpreted as a discretization of a parametric Ordinary…

Optimization and Control · Mathematics 2020-07-07 Joubine Aghili , Olga Mula

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the…

Methodology · Statistics 2026-04-08 Shuoxun Xu , Zijian Guo , Brooke R. Staveland , Robert T. Knight , Lexin Li

Neural Ordinary Differential Equations (ODE) are a promising approach to learn dynamic models from time-series data in science and engineering applications. This work aims at learning Neural ODE for stiff systems, which are usually raised…

Numerical Analysis · Mathematics 2021-10-04 Suyong Kim , Weiqi Ji , Sili Deng , Yingbo Ma , Christopher Rackauckas

The intersection of machine learning and dynamical systems has generated considerable interest recently. Neural Ordinary Differential Equations (NODEs) represent a rich overlap between these fields. In this paper, we develop a continuous…

Optimization and Control · Mathematics 2023-06-16 Maria Oprea , Mark Walth , Robert Stephany , Gabriella Torres Nothaft , Arnaldo Rodriguez-Gonzalez , William Clark

The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning…

Machine Learning · Statistics 2024-02-15 Yusuke Tanaka , Takaharu Yaguchi , Tomoharu Iwata , Naonori Ueda

The Path-dependent Neural Jump ODE (PD-NJ-ODE) is a model for online prediction of generic (possibly non-Markovian) stochastic processes with irregular (in time) and potentially incomplete (with respect to coordinates) observations. It is a…

Machine Learning · Statistics 2024-07-29 Florian Krach , Josef Teichmann

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

Machine Learning · Computer Science 2021-06-11 Sifan Wang , Paris Perdikaris

Deep models trained on large amounts of data often incorporate implicit biases present during training time. If later such a bias is discovered during inference or deployment, it is often necessary to acquire new data and retrain the model.…

Computer Vision and Pattern Recognition · Computer Science 2024-04-19 Niklas Penzel , Gideon Stein , Joachim Denzler