A probabilistic model for the numerical solution of initial value problems
Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely connected to the notion of inference on latent variables in statistics. We describe a class of algorithms that formulate the solution to an IVP as inference on a latent path that is a draw from a Gaussian process probability measure (or equivalently, the solution of a linear stochastic differential equation). We then show that certain members of this class are connected precisely to generalized linear methods for ODEs, a number of Runge--Kutta methods, and Nordsieck methods. This probabilistic formulation of classic methods is valuable in two ways: analytically, it highlights implicit prior assumptions favoring certain approximate solutions to the IVP over others, and gives a precise meaning to the old observation that these methods act like filters. Practically, it endows the classic solvers with `docking points' for notions of uncertainty and prior information about the initial value, the value of the ODE itself, and the solution of the problem.
Code (1)
Methods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics…
Bayesian Numerical Methods for Nonlinear Partial Differential Equations
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial …
Bayesian InferenceUncertainty QuantificationFenrir: Physics-Enhanced Regression for Initial Value Problems
We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of para…
Numerical Integrationparameter estimationregressionStable Implementation of Probabilistic ODE Solvers
Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by…
Pick-and-Mix Information Operators for Probabilistic ODE Solvers
Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic for…
Bayesian Inference