Accelerating Bayesian Inference over Nonlinear Differential Equations with Gaussian Processes

By Ben Calderhead, Mark A. Girolami, Neil D. Lawrence. 2008

220 citations recorded by the source corpus.

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AReS and MaRS - Adversarial and MMD-Minimizing Regression for SDEs (2019, methodology): “For this purpose, Calderhead et al. (2009) propose a product of experts heuristics that was accepted and reused until recently Wenk et al. (2018) showed that this heuristic leads to severe theoretical issues.”
Gradient Matching Methods for Computational Inference in Mechanistic Models for Systems Biology: A Review and Comparative Analysis (2015, methodology): “…for the state derivatives is then where Assuming additive Gaussian noise with a state-specific error variance γ i , from equation (2) we get (12) Calderhead et al. (2008), and Dondelinger et al. (2013) link the interpolant in equation (10) with the ODE model in equation (12) using a product of…”
Probabilistic ODE Solvers with Runge-Kutta Means (2014, background): “Hence, researchers in machine learning have repeatedly been interested in both the problem of inferring an ODE description from observed trajectories of a dynamical system [1, 2, 3, 4], and its dual, inferring a solution (a trajectory) for an ODE initial value problem (IVP) [5, 6, 7, 8].”
Zero Variance Differential Geometric Markov Chain Monte Carlo Algorithms (2014, methodology): “A population MCMC scheme can be employed to ensure convergence to the target distribution as suggested for example in Calderhead et al. (2009).”
Fast Gaussian Process Based Gradient Matching for Parameter Identification in Systems of Nonlinear ODEs (2018, methodology): “While the form of dynamical systems, the ODEs, can often be derived using expert knowledge, the parameters are usually unknown, can not be directly measured and have to be estimated from empirical time series data.”
Identifying Causal Structure in Large-Scale Kinetic Systems (2018, methodology): “[2007] suggest to regularize the smoother using the ODE itself, and Calderhead et al. [2009], Dondelinger et al. [2013], Wenk et al. [2018] use Gaussian Processes. Macdonald and Husmeier [2015] provide an overview of the different approaches based on gradient matching with a focus on parameter estim”
Learning Large Scale Ordinary Differential Equation Systems (2017, methodology): “Calderhead et al. (2009) demonstrated this in a Bayesian setting and proposed a collocation method based on Gaussian processes, Babtie et al. (2014) relied on a Gaussian process collocation based search of the model space similar to the EGM method that we have implemented, and Chen et al. (2016)…”
Efficient Sampling for Gaussian Process Inference using Control Variables (2008, methodology): “In such large systems where MCMC can become slow a combination of our method with the fast sampling scheme in [3] could be used to speed up the inference.”
Mechanistic Hierarchical Gaussian Processes (2014, background): “Other authors (Ramsay et al., 2007; Calderhead et al., 2009) develop novel techniques for estimating nonlinear differential equations utilizing smoothing based approximations.”
Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems (2016, background): “Recent developments include probabilistic solvers for linear systems [Hennig, 2015], differential equations [Calderhead et al., 2009, Dondelinger et al., 2013, Schober et al., 2014, Barber and Wang, 2014, Conrad et al., 2015, Macdonald et al., 2015, Owhadi, 2016, Kersting and Hennig, 2016] and…”