mlsanityData to 14 Oct 2026

Multiple shooting for training neural differential equations on time series

arXiv 2109.06786 ↗First posted 14 Sep 2021cs.LGcited by 17

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Abstract (arXiv)

Neural differential equations have recently emerged as a flexible data-driven/hybrid approach to model time-series data. This work experimentally demonstrates that if the data contains oscillations, then standard fitting of a neural differential equation may result in a flattened out trajectory that fails to describe the data. We then introduce the multiple shooting method and present successful demonstrations of this method for the fitting of a neural differential equation to two datasets (synthetic and experimental) that the standard approach fails to fit. Constraints introduced by multiple shooting can be satisfied using a penalty or augmented Lagrangian method.

Key facts. Cited by 17 papers in the map (data through 14 Oct 2026). Builds on 9 of its 32 references that are in the map. First posted on arXiv 14 Sep 2021.

Builds on

9 of 32 references.

References · 9

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2018On the Spectral Bias of Neural Networkscited by 1,341
  3. 2018A Tutorial on Bayesian Optimizationcited by 744
  4. 2019Augmented Neural ODEscited by 397
  5. 2018Regularisation of Neural Networks by Enforcing Lipschitz Continuitycited by 314
  6. 2018Training behavior of deep neural network in frequency domaincited by 174
  7. 2018Fashionable Modelling with Fluxcited by 53
  8. 2020Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flowscited by 37
  9. 2019DiffEqFlux.jl - A Julia Library for Neural Differential Equationscited by 33

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 4; 2023: 3; 2024: 0; 2025: 4; 2026: 6.

17 papers.

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  1. 2023Physics-Informed Machine Learning for Modeling and Control of Dynamical Systemscited by 28
  2. 2022Latent Neural ODEs with Sparse Bayesian Multiple Shootingcited by 17
  3. 2022Homotopy-based training of NeuralODEs for accurate dynamics discoverycited by 14
  4. 2022NeuralFMU: Presenting a workflow for integrating hybrid NeuralODEs into real world applicationscited by 3
  5. 2023CONFIDE: Contextual Finite Differences Modelling of PDEscited by 2
  6. 2023Effective Latent Differential Equation Models via Attention and Multiple Shootingcited by 2
  7. 2025Learning Physically Consistent Lagrangian Control Models Without Acceleration Measurementscited by 1
  8. 2022Learning Time Delay Systems with Neural Ordinary Differential Equationscited by 0
  9. 2025A condensing approach to multiple shooting neural ordinary differential equationcited by 0
  10. 2025Stable Robot Motions on Manifolds: Learning Lyapunov-Constrained Neural Manifold ODEscited by 0
  11. 2025CODE: A global approach to ODE dynamics learningcited by 0
  12. 2026Generative prediction of laser-induced rocket ignition with dynamic latent space representationscited by 0
5 more
  1. 2026MPINeuralODE: Multiple-Initial-Condition Physics-Informed Neural ODEs for Globally Consistent Dynamical System Learningcited by 0
  2. 2026Frequency-Domain Neural ODEs for Modeling Non-Linear Dynamical Systemscited by 0
  3. 2026Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategiescited by 0
  4. 2026TinyUDE: Solver-Free Universal Differential Equations on Microcontrollers via Lie-Taylor Jet Matchingcited by 0
  5. 2026Neural ODEs Meet Concurrent Learning: Stable Online Learning with Lyapunov Guaranteescited by 0