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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