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# Probabilistic Integration for Uncertainty Quantification in Differential Equation Models

Published on Jan 15, 20133856 Views

In this talk I discuss recent joint work with Oksana Chkrebtii, Prof. Dave Campbell and Prof. Mark Girolami, in which we develop a probabilistic formalism for solving systems of differential equations

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Probabilistic Integration for Uncertainty Quantification in Differential Equation Models00:00

Joint work with00:31

Differential equations are ubiquitous in many areas of Science and Engineering. (1)01:09

Differential equations are ubiquitous in many areas of Science and Engineering. (2)01:19

Differential equations are ubiquitous in many areas of Science and Engineering. (3)01:31

In many cases, particularly in biology, ...02:03

Often sparse, uncertain data with unobserved species03:04

We can adopt a Bayesian approach to characterise the uncertainty ...04:14

Nonlinear dynamics, correlation structure, identifiability...05:26

The challenge: How do we do this most efficiently?05:58

Note the assumption that the solution to the ODEs is exact!07:00

Majority of differential equations do not have analytical solutions and must be solved numerically ... (1)09:00

Majority of differential equations do not have analytical solutions and must be solved numerically ... (2)09:20

Main Idea (1)10:23

Main Idea (2)10:32

Why solve differential equations probabilistically?11:29

Recap12:39

In this work we shall model the derivative of the solution as a Gaussian process14:20

Goal15:52

Assuming Gaussian error model ...17:02

Solving ordinary differential equations is inherently sequential, and we propose the following scheme18:33

In practice, every time we iterate, the number of data points increases by one...19:50

To whet your appetite... in our paper, we ...20:21

An Example (1)21:08

An Example (2)21:55

Fully Probabilistic Inference22:46

Model parameters and initial conditions23:07

Auxiliary parameters - GP prior-precision and lengthscale.23:20

Extensions24:13

Conclusions26:20

Aknowledgements27:11