DiffEqUncertainty.jl is a component package in the DifferentialEquations ecosystem. It holds the utilities for solving uncertainty quantification. This includes quantifying uncertainties due to either:

• The propagation of initial condition and parametric uncertainties through an ODE
• The finite approximation of numerical solutions of ODEs and PDEs (ProbInts)

Here, we wish to compute the expected value for the number prey in the Lotka-Volterra model at 10s with uncertainty in the second initial condition and last model parameter. We will solve the expectation using two different algorithms, MonteCarlo and Koopman.

function f!(du,u,p,t)
    du[1] = p[1]*u[1] - p[2]*u[1]*u[2] #prey
    du[2] = -p[3]*u[2] + p[4]*u[1]*u[2] #predator
end

tspan = (0.0,10.0)
u0 = [1.0;1.0]
p = [1.5,1.0,3.0,1.0]
prob = ODEProblem(f,u0,tspan,p)

u0_dist = [1.0, Uniform(0.8, 1.1)]
p_dist = [1.5,1.0,3.0,truncated(Normal(1.0,.1),.6, 1.4)]

g(sol) = sol[1,end]

expectation(g, prob, u0_dist, p_dist, MonteCarlo(), Tsit5(); trajectories = 100_000)
expectation(g, prob, u0_dist, p_dist, Koopman(), Tsit5())  

If we wish to compute the variance, or 2nd central moment, of this same observable, we can do so as

centralmoment(2, g, prob, u0_dist, p_dist, Koopman(), Tsit5())[2]  

See SciMLTutorials.jl for additional examples.

DiffEqUncertainty.jl provides algorithms for computing the expectation of an observable, or quantity of interest, g of the states of a dynamical system as the system evolves in time. These algorithms are applicable to ODEs with initial condition and/or parametric uncertainty. Process noise is not currently supported.

You can compute the expectation by using the expectation function:

expectation(g, prob, u0, p, expalg, args...; kwargs...)

• g: A function for computing the observable from an ODE solution g(sol)
• prob: An ODEProblem
• u0: Initial conditions. This can include a mixture of Real and ContinuousUnivariateDistribution (from Distributions.jl) types, e.g. u0=[2.0, Uniform(1.0,2.0), Normal(4.0,1.0)]. This allows you to specify both uncertain and deterministic initial conditions
• p: ODE parameters. This also can include a mixture of Real and ContinuousUnivariateDistribution (from Distributions.jl) types.
• expalg: Expectation algorithm to use

The following algorithms are available:

• MonteCarlo: Provides a convenient wrapper to EnsembleProblem for computing expectations via Monte Carlo simulation. Requires setting trajectories >1. See the DifferentialEquations.jl documentation for additional details.
• Koopman: Leverages the Koopman operator to compute the expectation efficiently via quadrature methods. This capability is built on top of DifferntialEquations.jl and Quadrature.jl. See Quadrature.jl for additional options. For additional details on this algorithm, refer to The Koopman Expectation: An Operator Theoretic Method for Efficient Analysis and Optimization of Uncertain Hybrid Dynamical Systems

• quadalg: Quadrature algorithm. See Quadrature.jl for available algorithms
• maxiter: Maximum number of allowable quadrature iterations
• ireltol: Relative tolerance for quadrature integration
• iabstol: Absolute tolerance for quadrature integration
• nout: Output size of observable g. Used to specify vector-valued expectations
• batch: The preferred number of points to batch. This allows user-side parallelization of the expectation. See Quadrature.jl for additional details

These algorithms can also be used to compute higher order central moments via centralmoments. This function returns the central moments up to the requested number.

centralmoments(n, args...; kwargs...)

• n: highest-order central moment to be computed. centralmoments will return an n length array with central moments 1 through n
• args and kwargs: This function wraps expectation. See expectation for additional options.

Users interested in using this functionality should check out the DifferentialEquations.jl documentation.