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Extension functionality which uses Stan.jl, DynamicHMC.jl, and Turing.jl to estimate the parameters to differential equations and perform Bayesian probabilistic scientific machine learning

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This repository is a set of extension functionality for estimating the parameters of differential equations using Bayesian methods. It allows the choice of using CmdStan.jl, Turing.jl, DynamicHMC.jl and ApproxBayes.jl to perform a Bayesian estimation of a differential equation problem specified via the DifferentialEquations.jl interface.

To begin you first need to add this repository using the following command.

using DiffEqBayes

Tutorials and Documentation

For information on using the package, see the stable documentation. Use the in-development documentation for the version of the documentation, which contains the unreleased features.


using ParameterizedFunctions, OrdinaryDiffEq, RecursiveArrayTools, Distributions
f1 = @ode_def LotkaVolterra begin
 dx = a*x - x*y
 dy = -3*y + x*y
end a

p = [1.5]
u0 = [1.0,1.0]
tspan = (0.0,10.0)
prob1 = ODEProblem(f1,u0,tspan,p)

σ = 0.01                         # noise, fixed for now
t = collect(1.:10.)   # observation times
sol = solve(prob1,Tsit5())
priors = [Normal(1.5, 1)]
randomized = VectorOfArray([(sol(t[i]) + σ * randn(2)) for i in 1:length(t)])
data = convert(Array,randomized)

using CmdStan #required for using the Stan backend
bayesian_result_stan = stan_inference(prob1,t,data,priors)

bayesian_result_turing = turing_inference(prob1,Tsit5(),t,data,priors)

using DynamicHMC #required for DynamicHMC backend
bayesian_result_hmc = dynamichmc_inference(prob1, Tsit5(), t, data, priors)

bayesian_result_abc = abc_inference(prob1, Tsit5(), t, data, priors)

Using save_idxs to declare observables

You don't always have data for all of the variables of the model. In case of certain latent variables you can utilise the save_idxs kwarg to declare the oberved variables and run the inference using any of the backends as shown below.

 sol = solve(prob1,Tsit5(),save_idxs=[1])
 randomized = VectorOfArray([(sol(t[i]) + σ * randn(1)) for i in 1:length(t)])
 data = convert(Array,randomized)

 using CmdStan #required for using the Stan backend
 bayesian_result_stan = stan_inference(prob1,t,data,priors,save_idxs=[1])

 bayesian_result_turing = turing_inference(prob1,Tsit5(),t,data,priors,save_idxs=[1])
 using DynamicHMC #required for DynamicHMC backend
 bayesian_result_hmc = dynamichmc_inference(prob1,Tsit5(),t,data,priors,save_idxs = [1])

 bayesian_result_abc = abc_inference(prob1,Tsit5(),t,data,priors,save_idxs=[1])