mlsanityData to 7 Oct 2026

Variational multiple shooting for Bayesian ODEs with Gaussian processes

arXiv 2106.10905 ↗First posted 21 Jun 2021cs.LG stat.MLcited by 15

Not on the map. Why?

Abstract (arXiv)

Recent machine learning advances have proposed black-box estimation of unknown continuous-time system dynamics directly from data. However, earlier works are based on approximative ODE solutions or point estimates. We propose a novel Bayesian nonparametric model that uses Gaussian processes to infer posteriors of unknown ODE systems directly from data. We derive sparse variational inference with decoupled functional sampling to represent vector field posteriors. We also introduce a probabilistic shooting augmentation to enable efficient inference from arbitrarily long trajectories. The method demonstrates the benefit of computing vector field posteriors, with predictive uncertainty scores outperforming alternative methods on multiple ODE learning tasks.

Key facts. Cited by 15 papers in the map (data through 7 Oct 2026). Builds on 31 of its 68 references that are in the map. First posted on arXiv 21 Jun 2021.

Builds on

31 of 68 references.

References · 31

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2018Neural Ordinary Differential Equationscited by 3,733
  3. 2016Variational Inference: A Review for Statisticianscited by 1,466
  4. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  5. 2013Gaussian Processes for Big Datacited by 377
  6. 2017Stable Architectures for Deep Neural Networkscited by 340
  7. 2020Scalable Gradients for Stochastic Differential Equationscited by 264
  8. 2019Latent ODEs for Irregularly-Sampled Time Seriescited by 182
  9. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 168
  10. 2020Efficiently Sampling Functions from Gaussian Process Posteriorscited by 125
  11. 2020Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODEcited by 76
  12. 2018Probabilistic Recurrent State-Space Modelscited by 64
19 more
  1. 2018Learning unknown ODE models with Gaussian processescited by 54
  2. 2015MCMC for Variationally Sparse Gaussian Processescited by 44
  3. 2020Bayesian Neural Ordinary Differential Equationscited by 44
  4. 2014Variational Gaussian Process State-Space Modelscited by 43
  5. 2021MALI: A memory efficient and reverse accurate integrator for Neural ODEscited by 42
  6. 2017Identification of Gaussian Process State Space Modelscited by 31
  7. 2019Learning interpretable continuous-time models of latent stochastic dynamical systemscited by 29
  8. 2016Random Fourier Features for Operator-Valued Kernelscited by 22
  9. 2019Overcoming Mean-Field Approximations in Recurrent Gaussian Process Modelscited by 20
  10. 2021Differentiable Multiple Shooting Layerscited by 20
  11. 2019ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systemscited by 17
  12. 2020An Ode to an ODEcited by 17
  13. 2021Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy datacited by 13
  14. 2020Learning ODE Models with Qualitative Structure Using Gaussian Processescited by 9
  15. 2021Structure-preserving Gaussian Process Dynamicscited by 9
  16. 2019Monotonic Gaussian Process Flowcited by 5
  17. 2014Learning nonparametric differential equations with operator-valued kernels and gradient matchingcited by 4
  18. 2020Stochastic Differential Equations with Variational Wishart Diffusionscited by 4
  19. 2020A Modified Multiple Shooting Algorithm for Parameter Estimation in ODEs Using Adjoint Sensitivity Analysiscited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 6; 2024: 2; 2025: 2; 2026: 3.

15 papers.

Sort:

  1. 2022Latent Neural ODEs with Sparse Bayesian Multiple Shootingcited by 17
  2. 2023Modulated Neural ODEscited by 10
  3. 2023Free-Form Variational Inference for Gaussian Process State-Space Modelscited by 7
  4. 2023Bayesian Gaussian Process ODEs via Double Normalizing Flowscited by 7
  5. 2023Learning Energy Conserving Dynamics Efficiently with Hamiltonian Gaussian Processescited by 6
  6. 2023Learning Space-Time Continuous Neural PDEs from Partially Observed Statescited by 3
  7. 2024System-Aware Neural ODE Processes for Few-Shot Bayesian Optimizationcited by 3
  8. 2026Foundation Inference Models for Ordinary Differential Equationscited by 3
  9. 2022The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Modelscited by 2
  10. 2023Variational Gaussian Processes For Linear Inverse Problemscited by 2
  11. 2024Counterfactual-based Root Cause Analysis for Dynamical Systemscited by 2
  12. 2025Causal Structure Learning for Dynamical Systems with Theoretical Score Analysiscited by 2
3 more
  1. 2025CODE: A global approach to ODE dynamics learningcited by 0
  2. 2026Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processescited by 0
  3. 2026Function-Space Priors for Bayesian Neural ODEs with Application to Vessel Trajectory Predictioncited by 0