Ordinary Differential Equations

To avoid a frequent initial confusion for new readers, it may be helpful to point out that there are two common ways in which probabilistic methods are used in combination with ordinary differential equations: The “classic” problem of numerics is to infer the solution to an initial value problem given access to the differential equation. Below, we term this problem “solving ODEs”. The reverse problem, in some sense, has also found interest in machine learning: inferring a differential equation from (noisy) observations of trajectories that are assumed to be governed by this ODE. Below, this is listed under “inferring ODEs”.

Solving ODEs


  1. A role for symmetry in the Bayesian solution of differential equations
    Wang, Junyang, Cockayne, Jon, and Oates, Chris J
    Bayesian Analysis 2020


  1. Convergence Rates of Gaussian ODE Filters
    Kersting, H., Sullivan, T. J., and Hennig, P.
    ArXiv e-prints 2018
  2. Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration
    Abdulle, A., and Garegnani, G.
    ArXiv e-prints 2018
  3. Implicit Probabilistic Integrators for ODEs
    Teymur, Onur, Lie, Han Cheng, Sullivan, Tim, and Calderhead, Ben


  1. A probabilistic model for the numerical solution of initial value problems
    Schober, M., Särkkä, S., and Hennig, P.
    ArXiv e-prints 2016
  2. Active Uncertainty Calibration in Bayesian ODE Solvers
    Kersting, Hans P., and Hennig, Philipp
    In Uncertainty in Artificial Intelligence (UAI) 2016
  3. Probabilistic Linear Multistep Methods
    Teymur, Onur, Zygalakis, Kostas, and Calderhead, Ben


  1. A Random Riemannian Metric for Probabilistic Shortest-Path Tractography
    Hauberg, Søren, Schober, Michael, Liptrot, Matthew, Hennig, Philipp, and Feragen, Aasa
    In Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2015


  1. On solving Ordinary Differential Equations using Gaussian Processes
    Barber, D.
    ArXiv pre-print 1408.3807 2014
  2. Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics
    Hennig, Philipp, and Hauberg, Søren
    In Proc. of the 17th int. Conf. on Artificial Intelligence and Statistics (AISTATS) 2014
  3. Probabilistic shortest path tractography in DTI using Gaussian Process ODE solvers
    Schober, Michael, Kasenburg, Niklas, Feragen, Aasa, Hennig, Philipp, and Hauberg, Søren
    In Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014 2014
  4. Probabilistic ODE Solvers with Runge-Kutta Means
    Schober, Michael, Duvenaud, David K, and Hennig, Philipp


  1. Simulation of stochastic network dynamics via entropic matching
    Ramalho, Tiago, Selig, Marco, Gerland, Ulrich, and Enßlin, Torsten A.
    Phys. Rev. E 2013
  2. Bayesian Uncertainty Quantification for Differential Equations
    Chkrebtii, O., Campbell, D.A., Girolami, M.A., and Calderhead, B.
    Bayesin Analysis (discussion paper) 2013


  1. A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution
    Mosbach, Sebastian, and Turner, Amanda G.
    Computers & Mathematics with Applications 2009


  1. Solving noisy linear operator equations by Gaussian processes: Application to ordinary and partial differential equations
    Graepel, Thore
    In ICML 2003


  1. Bayesian solution of ordinary differential equations
    Skilling, J.
    Maximum Entropy and Bayesian Methods, Seattle 1991


  1. A Statistical Study Of The Accuracy Of Floating Point Number Systems
    Kuki, H., and Cody, W. J.
    Communications of the ACM 1973


  1. Test of Probabilistic Models for the Propagation of Roundoff Errors
    Hull, T. E., and Swenson, J. R.
    Communications of the ACM 1966

Inferring ODEs


  1. Scalable Variational Inference for Dynamical Systems
    Gorbach, Nico S, Bauer, Stefan, and Buhmann, Joachim M


  1. Gaussian Processes for Bayesian Estimation in Ordinary Differential Equations
    Wang, Yali, and Barber, David
    In International Conference on Machine Learning – ICML 2014


  1. Adaptive Markov chain Monte Carlo forward projection for statistical analysis in epidemic modelling of human papillomavirus
    Korostil, Igor A, Peters, Gareth W, Cornebise, Julien, and Regan, David G
    Statistics in medicine 2013


  1. Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes
    Calderhead, Ben, Girolami, Mark, and Lawrence, Neil D.